Chapter 13 & 14  Probability PART


 Jocelin Pitts
 2 years ago
 Views:
Transcription
1 Chapter 13 & 14  Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 1 / 91
2 Why Should We Learn Probability Theory? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 2 / 91
3 Let s Make a Deal! Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 3 / 91
4 Uncertainty Usually the results of a study, observational or experimental, are uncertain: if we repeat a study, we will not get exactly the same results. Example 1 (A coin). You toss a coin. Will it land heads or tails? Example 2 (A die). A regular die, a D6, is a cube with six faces: What number will a die show when it is rolled once? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 4 / 91
5 Uncertainty Example 3 (Box with marbles) A box contains 7 marbles: 2 red and 5 green. Each marble has an equal chance to be selected. One marble is to be drawn at random from the box. What color will it be? Example 4 (Box with tickets) A box contains 10 tickets labeled 1 through 10. What will be the number of a randomly selected ticket? Example 5 (Height of students) Seven students will be selected at random from the Math 148 class list and their heights will be measured. What will be the average height? Will it change if we choose different seven students? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 5 / 91
6 Probability What is the chance or probability that we will make the same conclusions every time when we replicate a study? What is the chance or probability that the histogram will change? Knowledge of probability theory will help us answer these questions. NOTE: In this part, the words chance and probability will have the same meaning. chance = probability Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 6 / 91
7 Inferential Statistics & Probability Theory In Parts II and III we used random samples to collect evidence and make inferences about population. Sample mean, x, estimates unknown population mean µ. Sample standard deviation, s, estimates unknown population standard deviation σ. Regression equation is a mathematical model which approximates the true relationship between x and y. In this part we will think in the opposite direction; we will reason from a known population to randomly selected samples. Probability Theory Inferential Statistics P opulation Sample Sample P opulation Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 7 / 91
8 Outcomes and Sample Space Probability theory deals with studies where the outcomes are not known for sure in advance. Usually, there are many possible outcomes for a study, we just do not know which particular outcome we will observe. Sample Space: The set of all possible outcomes of a study. The sample space of a study is denoted by S. Every repetition of a study, or a trial, produces a single outcome. Usually an outcome is computed from the values of the response variables. In Example 5 (Height of students) the outcome is the average height, ȳ, which is computed from the values of the response variable (y a student s height). Conclusions from a study are based on its outcome. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 8 / 91
9 Sample Spaces Example 1 (A coin) When you toss a coin, there are only two possible outcomes: a head or a tail. Then the sample space is S ={ head, tail} or simply S ={H, T}. Example 2 (A die) The sample space is S = {1, 2, 3, 4, 5, 6}. Example 3 (Box with marbles) We can draw either a red marble or a green one: S={R,G}. Example 4 (Box with tickets) S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 9 / 91
10 Sample Spaces Example 5 (Height of students) Of all our current examples, example 5 is the only continuous quantitative variable. The outcome of this study is ȳ, an average height. For any sample of 7 students the average height could be any number between 40 inches and 100 inches. Then the sample space is the interval: S = [40, 100]. Example 6 (Children) For every randomly chosen family the number of boys and girls is recorded. The outcomes in the sample space are pairs of the form (number of boys, number of girls) Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 10 / 91
11 Sample Space Example 6 (Two dice) Two dice are thrown. The sample space contains 36 outcomes as shown below: Mathematically, the sample space can be written as S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1),, (6, 6)}. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 11 / 91
12 Probability of Outcomes Probability: The theoretical probability of an outcome is the proportion (or percent) of times an outcome occurs in the sample space. Every outcome has a probability. If an experiment is repeated a large number of times, one would expect the ratio of occurrences of an outcome to number of experiments to be approximately the probability of that outcome. Notation: We will denote the probability of an outcome as where P( ) stands for probability. P(outcome), Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 12 / 91
13 Example 1 (A coin) A single coin flip has a simple sample space: S = {H, T }. There are only two potential outcomes: {H}, {T }. P(H) = number of H in sample space size of sample space = 1 2 P(T ) = number of H in sample space size of sample space = 1 2 Is it correct to assume that after flipping 10 coins, 5 would have come up heads? While imprisoned by the Germans during World War II, the South African statistician John Kerrich tossed a coin 10, 000times. Heads was the outcome 5,067 times! We have 67 more heads then expected. The difference between the observed percentage and the anticipated result is known as error. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 13 / 91
14 Example 1 (A coin) The graph shows the percentage of heads minus 50% versus the number of trials in Kerrich s experiment. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 14 / 91
15 Example 1 (A coin) From the graph we can see that the error approaches 0 as the number of tosses increases. After 10,000 tosses there are 67 extra heads, so the error is: % = 0.67%. 10, 000 It is already very close to 0! If we continue Kerrich s experiment, we can expect the error to become even smaller. We say the error converges to 0, which implies that the observed percentage of heads approaches 50% which suggests that P(H) = 0.5 has been calculated correctly. Question Why is the proportion of heads in Kerrich s experiment not EXACTLY equal to 0.5? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 15 / 91
16 Example 1 (A coin) Answer The number of heads is approximately half the number of tosses (5000), but it is off by 67 because of the chance error. The error 67 seems to be large in absolute units, but in relative units it is just 0.67%, which is a very small error. CONCLUSION: Kerrich s experiment supports the theoretical P(H) = 0.5. The observed number of heads may be interpreted as number of heads = half the number of tosses + chance error, where the error may appear to be large in absolute terms, but small relative to the number of tosses. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 16 / 91
17 Example (Blood Distribution) Data lifted from Wikipedia. The following Table shows the distribution of Blood Types of people in Australia. O+ A+ B+ AB+ O A B AB Total 40% 31% 8% 2% 9% 7% 2% 1% 100% The above distribution was obtained by computing the proportions of blood types for a huge representative group of Australian people. Therefore, we can say that the numbers in the table are the probabilities of different blood types for Australians. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 17 / 91
18 Example (Blood Distribution) 1 What is the probability of AB+ blood for Australians? Answer: P(AB+) = 0.02 If we randomly sample 1000 Australians, approximately 20 of them are expected to have AB+ blood type. However, due to chance variability, we will not observe exactly 20 people with an AB+ blood type. 2 In July 2009 the population of Australia was 28,395,716 people. Approximately, how many people in Australia have an A+ blood type? Answer: As the probability of having type A+ blood is 31%, the population, disregarding chance variability, of is , 395, 716 = 8, 802, , 802, 672 people Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 18 / 91
19 Example 2 (A die) An ace is the face of a die with one spot. Question What is the probability of getting an ace in a single roll? swer: number of faces with 1 P(ace) = = 1 number of faces 6 Indeed, a die has 6 faces and every face has the same chance to be observed in a single roll. Then P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1 6. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 19 / 91
20 Equally Likely Outcomes Assigning correct probabilities to individual outcomes often requires long observation of the random phenomenon. In some circumstances, however, we are willing to assume that individual outcomes are equally likely because of some balance in the phenomenon. For the equally likely outcomes: Probability of a single outcome = 1 Number of outcomes in S Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 20 / 91
21 Equally Likely Outcomes Ordinary coins have a physical balance that should make heads and tails equally likely. So the outcomes in Example 1 are equally likely: P(H) = P(T ) = 1 2 A fair die in Example 2 is equally likely to show any of its 6 faces in a single roll. P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1 6 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 21 / 91
22 Equally Likely Outcomes In Example 4, every one of the 10 tickets has the same chance to be drawn: P(1) = P(2) = P(3) = = P(8) = P(9) = P(10) = 1 10 In Example 6 (two dice), there are 36 equally likely outcomes. The probability of each outcome is In Example 3, the outcomes of drawing a red marble and drawing a green marble are not equally likely. There are more green marbles in the box, so the probability of selecting a green marble is greater. What is this probability? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 22 / 91
23 Computing Probabilities of Outcomes RULE: In problems involving drawing at random, the probability of getting an object of a particular type in a single draw is equal to proportion of objects of this type in the population. Example 3 (Box with marbles) The population is the box with marbles. The proportion of red marbles is 2/7, the proportion of green marbles is 5/7. This implies that in a single draw P(R) = 2 P(G) = Notice that P(R) + P(G) = 1. We will explore this phenomenon later in this chapter. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 23 / 91
24 Events We will give the definition of an event in words and mathematically. Definition of an event in words: An event is some statement about the random phenomenon. Events are usually denoted by capital letters as A, B, etc. Example of Events: (a) A = At least one head in two tosses of a coin, (b) B = Sum of the numbers on the dice is five, (c) C = A die rolled once shows an even number. Mathematical definition of an event: An event is a set of outcomes from the sample space S. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 24 / 91
25 Examples of Events (a) A = At least one head in two tosses of a coin. The sample space for this experiment is S = {HH, HT, TH, TT }. The event A = {HH, HT, TH} (b) B =Sum of the numbers on the dice is five. The sample space for this experiment contains 36 equally likely outcomes which are discussed in Example 2. The event B is B = {,,, } (c) C = A die rolled once shows an even number. The sample space is S = {1, 2, 3, 4, 5, 6}. The event C is C = {,, } Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 25 / 91
26 Occurrence of an Event We say that an event has occurred if ANY of the outcomes that constitute it occur. For instance, in Example (c) if we roll a die and it shows 4, we say that the event C has occurred. If the die shows 3, the event A does not occur because 3 is an odd number, so the event C does not contain this outcome. Let A be the event of landing at least one heads when tossing a coin 2 times, as in Example (a). If we toss and get {H, T } or {T, H} or {H, H} then we ll say that A has occurred. Only when we have two tails appearing does A not occur. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 26 / 91
27 Probabilities of Events The probability of an event A is denoted by P(A). In all our examples, except for Example 6, the sample space consisted of finitely many outcomes which we could list. In this case the sample space is called finite. An event is a collection of outcomes which support it. Theorem 1 (Probability of an event in a finite sample space) The probability of any event is the sum of probabilities of the outcomes making up the event. In the special case that the outcomes in the sample space are equally likely, the probability of an event E is computed as P(E) = # of outcomes supporting E Totat # of outcomes in S. (1) Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 27 / 91
28 Probabilities of Events (a) A = At least one head in two tosses of a coin. There are 4 equally likely outcomes in this experiment. The probability of every outcome is 1 4. Then P(A)= P(HH) + P(HT ) + P(TH)= = 3 4. (b) B = Sum of the numbers on the dice is five. There are 36 equally likely outcomes. Out of them, 4 outcomes support the event B. P(B) = 4 36 = 1 9. (c) C = A die rolled once shows an even number. There are 6 equally likely outcomes, 3 of them support C. P(C) = 3 6 = 1 2. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 28 / 91
29 Example 3 (Box with Marbles) A box contains 7 marbles, 2 red and 5 green. A marble is drawn at random, its color is recorded, and the marble is put back in the box. Then, the next marble is drawn at random. This example is of sampling with replacement, which we will define later. Find the probability that 2 red marbles are drawn. Solution: The sample space for this experiment is: S = {RR, RG, GR, GG} The outcomes are not equally likely since the red and green marbles have unequal chances to be selected. Intuitively, the RR is the least likely outcome. This outcome can be described: A = First two randomly chosen marbles are red Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 29 / 91
30 Example 3 (Box with Marbles) We can use the following two events to describe event A: A 1 = The first selected marble is red A 2 = The second selected marble is red Then it follows that A = A 1 and A 2. Note that if we replace the marble back to the box, a red marble has the same chance to be selected at every draw: P(A 1 ) = P(A 2 ) = 2 7 We have calculated P(A 1 ) and P(A 2 ), but how can we use this to calculate P(A 1 and A 2 )? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 30 / 91
31 Special Events We have the ability to calculate the probability of simple events; those resulting from one repetition of an experiment with equally likely outcomes. To tackle more complex problems, we must define some special events. Some special events: Case 1: Suretohappen or certain event. Case 2: Impossible event. Case 3: Opposite event. Case 4: Mutually exclusive or disjoint events. Case 5: Independent events. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 31 / 91
32 Certain Event Certain Event: An event which is guaranteed to happen at every repetition of the experiment. The event A = A head or a tail in a single toss of a coin. In every toss a coin lands up either heads or tails. There are no other possibilities. So A is a certain event. Notice that A = {H, T } which coincides with the sample space S for this experiment. The event B = rolling a die once will show a number less than 7, is a certain event. Notice that B = {1, 2, 3, 4, 5, 6} is the whole sample space. A certain event is equal to the sample space. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 32 / 91
33 Impossible Event Impossible Event: An event which never can occur. The event A = a coin lands up neither heads nor tails is an impossible event. In all the coinrelated experiment we usually assume that the coin may not lay on the edge. Eliminating this possibility makes event A impossible. The event B = the number that shows up when a die is rolled once is 2 is an impossible event. Mathematically an impossible event is written as, the empty set. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 33 / 91
34 Opposite Event Opposite Event: An event B is opposite to A if it happens whenever A does not happen. The event opposite to event A contains all the outcomes from the sample space S which do not belong to A. The opposite event for event A is called not A event. An opposite event for A is denoted as Ā. A and Ā split the sample space into two parts. (Also called a partition) Quite often, events are plotted on a graph as certain areas in the sample space. Such graphs are called the Venn diagrams. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 34 / 91
35 Opposite Event on a Venn diagram The Venn diagram shows the opposite event to the event A as the shaded region. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 35 / 91
36 Opposite Events: Examples (a) Event A = At least one head in two tosses of a coin. The opposite event Ā = It is NOT true that at least one H in two coin tosses = No heads in two tosses of a coin = {TT } The probability is given by P(Ā) = P(TT ) = 1 4 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 36 / 91
37 Opposite Events: Examples (b) Event B = Sum of five when rolling two dice. The opposite event: B = Sum of the numbers on the dice is NOT equal to five. B = {(1,1), (1,2), (1,3), (1,5),(1,6), (2,1), (2,2), (2,4), (2,5), (2,6), (3,1), (3,3), (3,4), (3,5), (3,6), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}. There are 32 equally likely outcomes in B. The sample space for this experiment contains 36 equally likely outcomes. This implies P( B) = = 8 9 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 37 / 91
38 Opposite Events: Examples (c) Event C = A die rolled once shows an even number. Event C can be described via its outcome space as C = {2, 4, 6}. The sample space for this experiment is S = {1, 2, 3, 4, 5, 6}. Then the event C will contain all the outcomes from S that C doesn t contain. In particular, C = {1, 3, 5} In words, C = A die rolled once shows an odd number. The probability P( C) = 1 2. Rule: For an event A, P(Ā) = 1 P(A). Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 38 / 91
39 Example 3 (Box with marbles) A box contains 7 marbles: 2 red and 5 green. Two marbles are chosen with replacement. We found the sample space S = {RR, RG, GR, GG}. We reconsider the event A =two red marbles in two draws with replacement = {RR}. What is the opposite event? It follows that Ā = {RG, GR, GG} can be expressed as: Ā = NOT exactly two red marbles in two draws = At most one red marble in two draws = Not all the marbles in two draws were red = At least one green marble in two draws P(Ā) = 3 4 = = 1 P(A) Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 39 / 91
40 Mutually Exclusive Events Mutually Exclusive Events: Two events A and B are mutually exclusive if they both cannot happen at the same time. Examples of mutually exclusive events: Events A = An odd numbered face showing and B = The 2 face showing are mutual events in a single die role. Events A = The coin lands up heads and B = The coin lands up tails are mutually exclusive events in a single coin flip experiment. Events A and B are mutually exclusive if they do not have common outcomes. We say A and B do not overlap or do not intersect. For this reason, mutually exclusive events are also called disjoint events. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 40 / 91
41 Mutually Exclusive Events A Venn diagram depicting two mutually exclusive events A and B in the sample space S. Single outcomes of an experiment which make up the sample space are always mutually exclusive. An event and its opposite are mutually exclusive: A and Ā are mutually exclusive. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 41 / 91
42 Example (Two dice) A pair of dice is rolled once. Consider the following events: A = Sum of numbers on the dice is 11. B = Both dice show an even number. C = Both dice show the same number. Are the events A and B mutually exclusive? Are the events B and C mutually exclusive? Let s express the events as sets of outcomes and see if they have common outcomes. A = { ; } B = { ; ; ; ; ; ; ; ; } C = { ; ; ; ; ; }. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 42 / 91
43 Example (Two dice) A = { ; } B = { ; ; ; ; ; ; ; ; } C = { ; ; ; ; ; }. Events A and B are mutually exclusive since there are no common outcomes. Events B and C are not mutually exclusive since they have three common outcomes. What are the probabilities of events A, B, and C? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 43 / 91
44 Intersection of Events If A and B are not mutually exclusive events, they have some common outcomes. The set of common outcomes is an event which is called the intersection of events A and B. We will denote the intersection of events A and B by A and B Example: B = { ; ; ; ; ; ; ; ; } C = { ; ; ; ; ; }. Denote by D the intersection of events B and C. Then B and C = D = { ; ; }. D = The face showing on each die are the same and even. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 44 / 91
45 Intersection of Events When we connect two events with AND, we are interested in their intersection. The intersection of two mutually exclusive events is the impossible event. If A and B are mutually exclusive, then A and B = Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 45 / 91
46 Example Suppose you toss a fair coin twice. You are counting heads, so two events of interest are A = First toss is a head, B = Second toss is a head. What is the probability of the intersection P(A and B)? The events A and B are both disjoint, they occur together when both tosses give heads. Indeed, the sample space in this experiment is S = {HH, HT, TH, TT } All the outcomes are equally likely. The events are A = {HH, HT } The intersection is A and B = {HH}. B = {HH, TH} P(A and B) = 1 4 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 46 / 91
47 Independent Events Independent Events: Two events A and B are independent of each other if knowing that one event occurs does not change the probability that the other event occurs. Tosses of a fair coin are independent events. Further, it is equally likely that each side show after a toss. Whether there are two heads in a row or twenty, the chance of getting a head next time is still 0.5. It is the memoryless property of the coin. A fair die also has the memoryless property, i.e., the rolls are independent. For instance, the probability of an ace at every roll is 1 6 no matter how many aces has appeared in the previous rolls. We will assume that child births are independent events. Child births are independent events. The actual probability of a boy is slightly higher than 0.5, but we will assume it to be 0.5. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 47 / 91
48 Independent Events Consider two events arising from rolling two dice: A = Getting a 6 on the first roll. B = The sum of the numbers seen on the first and second rolls is 11 Are A and B independent events? Solution 1: The event B consists of the following outcomes: B = { ; }. The probability of event B is P(B) = 2 36 = Suppose event A has happened. Then the possible outcomes for the second trial are 1, 2, 3, 4, 5, 6. Event B will happen if the second outcome was 5. The probability of this is 1 6 Knowing that event A has happened changes the probability that event B will happen. Hence, A and B are not independent events. which is larger than Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 48 / 91
49 Independent and Disjoint Events If events A and B are independent, they can not be mutually exclusive. Independent events always have common outcomes (intersect). The opposite statement is also true: Mutually exclusive events are not independent (dependent). Explanation: If we know that A and B are mutually exclusive events with nonzero probabilities P(A) > 0 and P(B) > 0, and we know that event A has happened, this implies that the probability that event B also happened is 0. Example: Toss a coin twice. Two events A = First outcome is a head and B = Second outcome is a head are independent, but not mutually exclusive. A = {HH, HT }, B = {TH, HH} Events A and B are not mutually exclusive since they have a common outcome {HH}. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 49 / 91
50 Union of Events Union: The union of events A and B is the event which happens when either event A or event B or both happen. The union of two events is expressed as A or B In mathematics, the word or is not inclusive, rather it is exclusive. Inclusive means either or, but not both; as in a restaurant. Exclusive means either or both. In other words, the union of events A and B is the event which happens when at least one of events A or B happens. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 50 / 91
51 Union of Events: Venn diagram Mathematically, the union of two events A and B is found by combining the outcomes from A and B into a single set A or B. The following Venn diagram illustrates the union of events A and B. The shaded region corresponds to A or B. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 51 / 91
52 Example Find the probability of event E = at least one 3 in two successive rolls of a die. Solution: The event E consists of the following outcomes: ; ; ; ; ; ; ; ; ; ; The sample size for the two dice problem has 36 equally likely outcomes. As a consequence: P(E) = Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 52 / 91
53 Example Observe that event E can be represented as the union of two events: A = First outcome is 3 = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)} B = Second outcome is 3 = {(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3)} Therefore: E = At least one 3 in two successive rolls of a die = First outcome is 3 or second outcome is 3 = A or B When combining the outcomes from two events into a single event, do not repeat the same outcomes twice. Observe that each of the events A and B has 6 outcomes, but their union (event E) has 11 outcomes. This is because A and B have one common outcome (3, 3) which is written just once in E. Notice A and B= {(3, 3)}. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 53 / 91
54 Rules of Probability The following rules simplify many probability computations: Rule 1: The probability P(A) of any event A satisfies 0 P(A) 1. In other words, the probability of any event is between 0 and 1. Rule 2: If S is the sample space for an experiment, then P(S) = 1. The probability of a certain event is 1. Rule 3: The probability of an impossible event is 0. Hence, P( ) = 0. Rule 4: If events A and B are independent, then the probability that they both happen is the product of their probabilities: P(A and B) = P(A) P(B) Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 54 / 91
55 Rules of Probability Rule 5: For any events A and B, the probability of their union is equal to the sum of their individual probabilities minus the probability of their intersection: P(A or B) = P(A) + P(B) P(A and B) Subtracting the probability of the intersection is needed to avoid double counting. A special case: if A and B are disjoint events, then the probability of their union is the sum of individual probabilities: Rule 6: For any event A P(A or B) = P(A) + P(B) P(Ā) = 1 P(A) Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 55 / 91
56 A Few Remarks Some important notes about the rules of probability: 1. The closer the probability of an event is to 1, the more likely this event is to happen. 2. Unlikely events have probabilities close to Outcomes partition the sample space: Every cell in the above picture is an outcome. 4. Probabilities of all the outcomes add up to 1 because the set of all outcomes is S and P(S) = 1. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 56 / 91
57 Example We reconsider the following events arising from rolling a die twice: E = At least one 3 in two rolls of a die A = The first outcome is 3 B = The second outcome is 3. It should be clear that E = A or B, or that E is the union of both A and B. The experiment which consists in throwing 2 fair dice has 36 equally likely outcomes. P(A) = 6 36 = 1 6 = P(B) The intersection of events A and B is { P(A and B) = } with probability P(E) = P(A or B)= P(A) + P(B) P(A and B)= = Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 57 / 91
58 Example Problem 1. A total of 30% of American males smoke cigarettes, 7% smoke cigars, and 5% smoke both cigars and cigarettes. What percentage of males smoke neither cigars nor cigarettes? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 58 / 91
59 Example (Blood Type  2) From Moore and McCabe All human blood can be ABOtyped as one of A, B, O, or AB, but the distribution of the types varies among groups of people. Here is the distribution of blood types for a randomly chosen person in the US: Blood type O A B AB U.S. probability ? (a) What is the probability of type AB blood in the US? P(AB) = = 0.04 (b) Maria has type B blood. She can safely receive blood transfusions from people with blood types O and B. What is the probability that a randomly chosen American can donate blood to Maria? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 59 / 91
60 Example (Blood Type  2) Consider the events: Blood type O A B AB U.S. probability O = Randomly selected person has O blood type B = Randomly selected person has B blood type. Events O and B are disjoint since a person can not have type O and type B of blood at the same time. We are interested in the probability that a randomly selected person has a blood type of either O or B type. Then P(O or B)= P(O) + P(B) = = Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 60 / 91
61 Example (Blood Type  3) The probability of randomly choosing an individual with blood type A is 0.43 in the US and 0.22 in China. What is the probability of randomly and independently choosing two people, one from the US and the other from China, that both have blood type A? Solution: Define the following events: X = An American has A blood type Y = A Chinese has A blood type Notice that events X and Y are independent. We are interested in the probability of the event X and Y. By the multiplication rule for independent events P(E and F ) = P(E) P(F ) = = Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 61 / 91
62 Extending the Rules of Probability 1 The multiplication rule for a collection of m mutually independent events. For any collection of events A 1, A 2,..., A m that are mutually independent, the probability of the intersection of all the events is equal to the product of their individual probabilities: P(A 1 and A 2 and... and A m ) = P(A 1 ) P(A 2 )... P(A m ). 2 The addition rule for m disjoint events. If events A 1, A 2,..., A m are disjoint in the sense that they do not have common outcomes, then P(A 1 or A 2 or... or A m ) = P(A 1 ) + P(A 2 ) + + P(A m ). Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 62 / 91
63 Example (Universal Donors) From Moore and McCabe People with Onegative blood are universal donors. That is, any patient can receive a transfusion of Onegative blood. Only 7% of the American population have Onegative blood. If 3 people appear at random to give blood, what is the probability that at least one of them is a universal donor? Solution: Let s code the outcomes of the experiment in the following way: Y means a randomly selected person has Onegative blood N means a person does NOT have Onegative blood. Then the sample space for this experiment is as follows: S = {NNN, NNY, NYN, YNN, NYY, YNY, YYN, YYY } Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 63 / 91
64 Example (Universal Donors) We are given that P(Y ) = Notice that Y and N are opposite events! Therefore P(N) = 1 P(Y ) = = We are interested in the probability of the following event: A = at least one of the three people is a universal donor = at least one Y in the outcome = { NNY, NYN, YNN, NYY, YNY, YYN, YYY} A has all the outcomes from S except {NNN}, which should be the opposite event. Ā = None of the three people is a universal donor = No Y in the outcome = {NNN} Clearly it ll be easier to find the probability of Ā rather than A! Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 64 / 91
65 Example (Universal Donors) S = {NNN, NNY, NYN, YNN, NYY, YNY, YYN, YYY } A = {NNY, NYN, YNN, NYY, YNY, YYN, YYY } Ā = {NNN} Recall that each outcome is the result of three independent events. Therefore, we may use the multiplication rule of mutually independent events: P(Ā) = P(N and N and N) = P(N) P(N) P(N) = (0.93) 3 = As Ā is opposite A we have P(A) = 1 P(Ā) = = Notice that P(A) can be calculated without finding P(Ā) by using multiplication and addition formulas: P(A) = P(NNY or NYN or YNN or NYY or YNY or YYN or YYY ) = P(N)P(N)P(Y ) + P(N)P(Y )P(N) + P(Y )P(N)P(N) +... Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 65 / 91
66 Two Sampling Schemes We will consider two important sampling schemes: Sampling with Replacement Sampling without Replacement We explore these sampling techniques through the following problem: Problem. A box contains r red and g green marbles. Define the following events: R i = The marble selected on the ith draw is red. For example, R 1 is the event for which the first drawn marble is red. We are interested in the probabilities of events R 1, R 2, R 3,.... Since every marble has the same chance to be picked, we see that P(R 1 ) = r r + g Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 66 / 91
67 Two Sampling Schemes What are the probabilities of events R 2, R 3,...? That depends on the sampling scheme: Sampling with replacement: Suppose we return the marble to the box after each draw. In this case the conditions of experiment do not change since each subsequent draw is made from the same population. 1 Subsequent draws are independent. 2 The probability to select a red marble is the same at every draw: P(R i ) = r, i = 1, 2,... r + g Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 67 / 91
68 Two Sampling Schemes Sampling without replacement: After each draw the drawn marble is not returned to the box. Subsequently, conditions of the experiment change from draw to draw because the population of marbles in the box changes after each draw. 1 The probability of selecting a red marble changes with each draw! 2 Subsequent draws are NOT independent. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 68 / 91
69 Example (Box with Marbles) A box contains 7 marbles  2 red and 5 green. We have considered the event A: A = The first two chosen marbles are red. We used the following two events to describe event A : We found that A = A 1 and A 2 A 1 = The first selected marble is red, A 2 = The second selected marble is red. We know A 1 but we are not sure how to calculate A 2 ; is the sampling done with or without replacement? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 69 / 91
70 Example (Box with Marbles) Sampling with replacement: Subsequent draws are independent. P(A 1 ) = P(A 2 ) = 2 7 P(A) = P(A 1 ) P(A 2 ) = = 4 49 Sampling without replacement: The probability of A 1 is still 2 7. The probability of A 2 depends on the outcome of the first draw: The probability to select a red marble on the second draw, given that the first selected marble was red is 1 6. The probability to select a red marble given that the first selected marble was green is 2 6. Sampling multiple times without replacement is not independent! The multiplication rule for independent events cannot be used! Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 70 / 91
71 Conditional Probability Conditional Probability: Let A and B be events. If the events are not independent, then the occurrence of B alters the probability that A will occur. The conditional probability of event A given that event B has happened is denoted P(A B). Example: (Single roll of a fair die) Consider two events: A = The die shows 3 B = The die shows odd. The ordinary, unconditional, probabilities of events A and B are P(A) = 1 6 P(B) = 1 2 Suppose we know that event B happened. Then the conditional probability of A given B is P(A B) = 1 3 since there are only 3 odd outcomes possible: 1, 3, and 5, all equally likely. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 71 / 91
72 Conditional Probability The conditional probability of an event A given an event B is P(A B) = P(A and B) P(B) A B S When we say given B ; we are only considering outcomes covered by event B. Ignore everything outside of B! Then find the probability that an A outcome lies in B. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 72 / 91
73 Conditional Probability: Examples (a) Two machines, I and II, produce bolts. Five percent of those from I and ten percent of those from II are defective. This can be written as P(defective machine I) = 0.05 P(defective machine II) = Notice that we do not know P(defective), nor can we calculate it with the information supplied. (b) Suppose a mortality table shows that the probability of dying within one year for a 25yearold male is This can be stated as P(male dying within one year age 25) = Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 73 / 91
74 Conditional Probability Note 1: Knowledge that B has occurred effectively reduces the sample space from S to B. This is sometimes called the reduced sample space. Therefore, when interpreting the area of an event on the Venn diagram as its probability, P(A B) is the proportion of the area of B occupied by A. Note 2: If events A and B are mutually exclusive, then P(A B) = 0 as P(A and B) = 0. Note 3. Recall the formula: P(A B) = P(A and B) P(B) With algebraic manipulation we find: P(A and B) = P(A B) P(B) This formula is the general multiplication rule. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 74 / 91
75 The General Multiplication Rule As and is commutative, P(A B) P(B) = P(A and B) = P(A and B) = P(B A) P(A) The order of conditioning may be changed if needed. Extension of the General Multiplication Rule: The events A 1, A 2,..., A n are not necessarily independent. P(A 1 and A 2 and A 3 and A 4 and...) = = P(A 1 ) P(A 2 A 1 ) P(A 3 A 1 and A 2 ) P(A 4 A 1 and A 2 and A 3 )... Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 75 / 91
76 Intersection of 3 Events on a Venn Diagram The intersection of three events A, B, and C has probability P(A and B and C) = P(A) P(B A) P(C A and B) A B C S Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 76 / 91
77 Example (Athletes) (From Moore and McCabe) Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these, only 1.7% enter major league professional sports. About 40% of the athletes who compete in college and then reach the pros have a career of more than 3 years. Define these events A = Competes in college, B = Competes professionally, C = Professional career longer than 3 years. What is the probability that a high school male athlete competes in college, reaches a professional level and then goes on to have a pro career of more than 3 years? Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 77 / 91
78 Example (Athletes) We are given that The probability we want is, therefore, P(A) = 0.05 P(B A) = P(C A and B) = 0.4. P(A and B and C) = P(A) P(B A) P(C A and B) = = Interpretation: Only about 3 of every 10,000 high school male athletes can expect to compete in college, reach a professional level and have a professional career for more than 3 years. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 78 / 91
79 Conditional Probability and Independence When events A and B are independent, knowing that event B has occurred gives no additional information about the occurrence of event A. This can be expressed as P(A B) = P(A) To check whether two events A and B are independent, you should check either equality: P(A B) = P(A) or P(A and B) = P(A) P(B). Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 79 / 91
80 Example (Degrees) (From Moore and McCabe) The counts (in thousands) of earned degrees in the US in the academic year, classified by level and by the sex of the degree recipient: Bachelor s Master s Professional Doctorate Total Female Male Total Tables of this type are called crosstabulation or contingency tables. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 80 / 91
81 Example (Degrees) (a) If you choose a degree recipient at random, what is the probability that the person that you choose is a woman? P(W ) = 1119 thousands 1944 thousands (b) What is the probability of choosing a woman, given that the person chosen received a professional degree? P(W P) = (c) Are the events choose a woman and choose a professional degree recipient independent? Why or why not? We found in (a) that P(W ) = In (b) we found P(W P) = Since P(W ) P(W P), the events are not independent. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 81 / 91
82 Example (Degrees) (d) A randomly chosen person is a man. What is the probability that he received a bachelor s degree? P(B M) = (e) Use the multiplication rule to find the probability of choosing a male bachelor s recipient. Check your result by finding this probability directly from the table of counts. P(M and B) = P(B M) P(M) = = , which is the probability obtained directly from the table of counts. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 82 / 91
83 Example (Box with Marbles) Recall that there are 2 red and 5 green marbles in a box. We defined the following events: We found A = A 1 and A 2, and A = Two randomly chosen marbles are red A 1 = First selected marble is red A 2 = Second selected marble is red. P(A 1 ) = P(A 2 ) = 2 7. We want to compute the probability of A under sampling without replacement: P(A) = P(A 1 and A 2 ) = P(A 2 A 1 )P(A 1 ) = = We previously calculated that the probability of event A under sampling with replacement is Since 4 49 > 1 21, event A is more likely to occur under sampling with replacement. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 83 / 91
84 Examples Example: Two birds are selected at random without replacement from a cage containing five male and two female finches. What is the probability that both are males? P(M and M) = P(M) P(M M) = = Example: A large basket of fruit contains 3 oranges, 2 apples and 5 bananas. If two fruit are chosen at random without replacement, what is the probability that one of the selected fruits is an apple and the other one is an orange? P(A and O) = P(A) P(O A) = = 1 15 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 84 / 91
85 Examples Example: A box of 10 items has 2 defective items. Three items are selected by the researcher without replacement. (a) What is the probability that the researcher will obtain no defective items? P( D and D and D) = P( D) P( D D) P( D D and D) = = 7 15 (b) Given that the researcher finds the first item selected as defective, what is the probability that the researcher will also have the other defective item? P( ( D and D) D) = P( D D) P(D D and D) = = 1 9 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 85 / 91
86 Example (Roulette) Example 20 (Roulette), from Moore and McCabe A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. (a) What is the probability that the ball will land at any one slot? P(any slot) = 1 38 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 86 / 91
87 Example (Roulette) (b) If you bet on red, you will win if the ball lands on a red slot. What is the probability of winning? P(red) = # of red slots 38 = = 9 19 < 1 2 Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 87 / 91
88 Example (Roulette) (c) The slot numbers are laid on a board on which gamblers place their bets. One column of numbers on the board contains all multiples of 3, i.e., 3, 6, 9,..., 36. You place a column bet that wins if any of these numbers comes up. What is you probability of winning? P(Slot # is a multiple of 3) = = 6 19 < 13. In fact, every game in a casino offers options to the gambler that has less than a 50% chance of winning. In later chapters we will discuss probability and payout. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 88 / 91
89 Example (Lottery) (From Moore and McCabe.) A state lottery s Pick 3 game asks players to choose a threedigit number, 000 to 999. The state chooses the winning threedigit number at random. You win if the winning number contains the digits in you number, in any order. (a) Your number is 123. What is your probability of winning? (b) Your number is 112. What is your probability of winning? Solution: First of all, note that the sample space S for the experiment, which consists in choosing a 3digit number at random, contains 1000 equally likely outcomes: S = {000, 001,..., 999}. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 89 / 91
90 Example (Lottery) (a) Consider the event A: a chosen number has digits 1, 2, and 3. A = {123, 132, 213, 231, 312, 321} Then P(A) = = 0.006, quite a small chance! (b) Define the event B: a chosen number has two 1 s and one 2. B = {112, 121, 211} Then P(B) = = You do not want the digits to repeat in your lottery ticket! Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 90 / 91
91 Concluding Remarks Observation 1: and and means that we are looking at the intersection of events. We need to compute the probability that ALL events in the statement occur simultaneously. Probability computations in this case usually involve using one of the multiplication rules. Observation 2: or or means that we are looking at the union of events. We need to compute the probability that at least one event in the union occurs. Probability computations in this case usually involve either one of the addition rules or the rule for opposite events. Observation 3: at least, at most, not exactly, not equal Most of the problems of this type reduce to computing probabilities of the unions of events. It is often easier to compute the probability of the opposite event first. Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 91 / 91
Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then
Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance
More informationBasic concepts in probability. Sue Gordon
Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are
More informationSection 65 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationMath 421: Probability and Statistics I Note Set 2
Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the
More informationProbability. Experiment  any happening for which the result is uncertain. Outcome the possible result of the experiment
Probability Definitions: Experiment  any happening for which the result is uncertain Outcome the possible result of the experiment Sample space the set of all possible outcomes of the experiment Event
More informationAP Stats  Probability Review
AP Stats  Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationWorked examples Basic Concepts of Probability Theory
Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationMATH 3070 Introduction to Probability and Statistics Lecture notes Probability
Objectives: MATH 3070 Introduction to Probability and Statistics Lecture notes Probability 1. Learn the basic concepts of probability 2. Learn the basic vocabulary for probability 3. Identify the sample
More informationUnit 19: Probability Models
Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,
More informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
More information+ Section 6.2 and 6.3
Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities
More informationChapter 4 Probability
The Big Picture of Statistics Chapter 4 Probability Section 42: Fundamentals Section 43: Addition Rule Sections 44, 45: Multiplication Rule Section 47: Counting (next time) 2 What is probability?
More informationMATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics
MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should
More informationPROBABILITY. Chapter Overview Conditional Probability
PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the
More informationBasic Probability Theory I
A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population
More informationMath 140 Introductory Statistics
The Big Picture of Statistics Math 140 Introductory Statistics Lecture 10 Introducing Probability Chapter 10 2 Example Suppose that we are interested in estimating the percentage of U.S. adults who favor
More informationProbability. Vocabulary
MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationLesson 1: Experimental and Theoretical Probability
Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.
More informationProbability OPRE 6301
Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.
More informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More information7.1 Sample space, events, probability
7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More information36 Odds, Expected Value, and Conditional Probability
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
More informationMAT 1000. Mathematics in Today's World
MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities
More informationIn the situations that we will encounter, we may generally calculate the probability of an event
What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead
More informationProbability and Venn diagrams UNCORRECTED PAGE PROOFS
Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve
More informationProbability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.
1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event
More informationOdds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes.
MATH 11008: Odds and Expected Value Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes. Suppose all outcomes in a sample space are equally likely where a of them
More informationSolution (Done in class)
MATH 115 CHAPTER 4 HOMEWORK Sections 4.14.2 N. PSOMAS 4.6 Winning at craps. The game of craps starts with a comeout roll where the shooter rolls a pair of dice. If the total is 7 or 11, the shooter wins
More informationProbability. Experiment is a process that results in an observation that cannot be determined
Probability Experiment is a process that results in an observation that cannot be determined with certainty in advance of the experiment. Each observation is called an outcome or a sample point which may
More information4.19 What s wrong? Solution 4.25 Distribution of blood types. Solution:
4.19 What s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer. a) If two events are disjoint, we can multiply their probabilities
More informationChapter 5: Probability: What are the Chances? Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation
Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 5 Probability: What Are
More informationBasic Probability. Probability: The part of Mathematics devoted to quantify uncertainty
AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.
More informationMATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS
MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution
More informationAn event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event
An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the
More informationEvents. Independence. Coin Tossing. Random Phenomena
Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,
More informationV. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE
V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationSection 2.1. Tree Diagrams
Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.
More informationConditional Probability
Conditional Probability We are given the following data about a basket of fruit: rotten not rotten total apple 3 6 9 orange 2 4 6 total 5 10 15 We can find the probability that a fruit is an apple, P (A)
More informationIn this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events.
Lecture#4 Chapter 4: Probability In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. 42 Fundamentals Definitions:
More informationA Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability  1. Probability and Counting Rules
Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More information1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.
1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first
More information8.5 Probability Distributions; Expected Value
Math 07  Finite Math.5 Probability Distributions; Expected Value In this section, we shall see that the expected value of a probability distribution is a type of average. A probability distribution depends
More informationSection 8.1 Properties of Probability
Section 8. Properties of Probability Section 8. Properties of Probability A probability is a function that assigns a value between 0 and to an event, describing the likelihood of that event happening.
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationPROBABILITY NOTIONS. Summary. 1. Random experiment
PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non
More informationAssigning Probabilities
What is a Probability? Probabilities are numbers between 0 and 1 that indicate the likelihood of an event. Generally, the statement that the probability of hitting a target that is being fired at is
More informationBasic Probability Theory II
RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample
More informationChapter 5  Probability
Chapter 5  Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set
More informationExample. For example, if we roll a die
3 Probability A random experiment has an unknown outcome, but a well defined set of possible outcomes S. The set S is called the sample set. An element of the sample set S is called a sample point (elementary
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationThe concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics.
Elementary probability theory The concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics. 1.1 Experiments, outcomes and sample
More informationMath 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event
Math 1320 Chapter Seven Pack Section 7.1 Sample Spaces and Events Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment
More informationProbability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.
Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationProbability Review. ICPSR Applied Bayesian Modeling
Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How
More informationChapter 4: Probabilities and Proportions
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,
More informationTopic 6: Conditional Probability and Independence
Topic 6: September 1520, 2011 One of the most important concepts in the theory of probability is based on the question: How do we modify the probability of an event in light of the fact that something
More informationDistributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment
C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning
More information34 Probability and Counting Techniques
34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.
More information**Chance behavior is in the short run but has a regular and predictable pattern in the long run. This is the basis for the idea of probability.
AP Statistics Chapter 5 Notes 5.1 Randomness, Probability,and Simulation In tennis, a coin toss is used to decide which player will serve first. Many other sports use this method because it seems like
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationPROBABILITY. Chapter Overview
Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability
More informationNotes 3 Autumn Independence. Two events A and B are said to be independent if
MAS 108 Probability I Notes 3 Autumn 2005 Independence Two events A and B are said to be independent if P(A B) = P(A) P(B). This is the definition of independence of events. If you are asked in an exam
More informationCHAPTER 3: PROBABILITY TOPICS
CHAPTER 3: PROBABILITY TOPICS Exercise 1. In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities
More informationSample Space, Events, and PROBABILITY
Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationLecture 11: Probability models
Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample
More informationH + T = 1. p(h + T) = p(h) x p(t)
Probability and Statistics Random Chance A tossed penny can land either heads up or tails up. These are mutually exclusive events, i.e. if the coin lands heads up, it cannot also land tails up on the same
More information7.5 Conditional Probability; Independent Events
7.5 Conditional Probability; Independent Events Conditional Probability Example 1. Suppose there are two boxes, A and B containing some red and blue stones. The following table gives the number of stones
More informationQuestion: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationStatistics 1040 Summer 2009 Exam III NAME. Point score Curved Score
Statistics 1040 Summer 2009 Exam III NAME Point score Curved Score Each question is worth 10 points. There are 12 questions, so a total of 120 points is possible. No credit will be given unless your answer
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 4.2 Homework Answers 4.17 Choose a young adult (age 25 to 34 years) at random. The probability is 0.12 that the person chosen
More informationWhat is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts
Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called
More informationExample: If we roll a dice and flip a coin, how many outcomes are possible?
12.5 Tree Diagrams Sample space Sample point Counting principle Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible
More informationRELATIONS AND FUNCTIONS
008 RELATIONS AND FUNCTIONS Concept 9: Graphs of some functions Graphs of constant function: Let k be a fixed real number. Then a function f(x) given by f ( x) = k for all x R is called a constant function.
More informationProbability and Counting
Probability and Counting Basic Counting Principles Permutations and Combinations Sample Spaces, Events, Probability Union, Intersection, Complements; Odds Conditional Probability, Independence Bayes Formula
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationPROBABILITY. Chapter. 0009T_c04_133192.qxd 06/03/03 19:53 Page 133
0009T_c04_133192.qxd 06/03/03 19:53 Page 133 Chapter 4 PROBABILITY Please stand up in front of the class and give your oral report on describing data using statistical methods. Does this request to speak
More information7.S.8 Interpret data to provide the basis for predictions and to establish
7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an
More informationAccording to the Book of Odds, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61.
Probability Law of large numbers: if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single number. Probability: probability
More informationThe overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES
INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number
More informationChapter 5 Section 2 day 1 2014f.notebook. November 17, 2014. Honors Statistics
Chapter 5 Section 2 day 1 2014f.notebook November 17, 2014 Honors Statistics Monday November 17, 2014 1 1. Welcome to class Daily Agenda 2. Please find folder and take your seat. 3. Review Homework C5#3
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationSection 6.2 ~ Basics of Probability. Introduction to Probability and Statistics SPRING 2016
Section 6.2 ~ Basics of Probability Introduction to Probability and Statistics SPRING 2016 Objective After this section you will know how to find probabilities using theoretical and relative frequency
More informationProbability: Events and Probabilities
Probability: Events and Probabilities PROBABILITY: longrun relative frequency; likelihood or chance that an outcome will happen. A probability is a number between 0 and 1, inclusive, EVENT: An outcome
More informationBasics of Probability
Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.
More informationMath 141. Lecture 1: Conditional Probability. Albyn Jones 1. 1 Library jones/courses/141
Math 141 Lecture 1: Conditional Probability Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Definitions: Sample Space, Events Last Time Definitions: Sample Space,
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More informationThis is Basic Concepts of Probability, chapter 3 from the book Beginning Statistics (index.html) (v. 1.0).
This is Basic Concepts of Probability, chapter 3 from the book Beginning Statistics (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More information