# Chapter 4 Probability

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability? Probability is a mathematical description of randomness and uncertainty. Random experiment is an experiment that produces an outcome that cannot be predicted in advance (hence the uncertainty). : coin toss The result of any single coin toss is random. Possible outcomes: Heads (H) Tails (T) Coin toss The result of any single coin toss is random. But the result over many tosses is predictable. The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials. The Law of Large Numbers As a procedure repeated again and again, the relative frequency probability of an event tends to approach the actual probability. Spinning a coin First series of tosses Second series 1

2 Sample Space This list of possible outcomes an a random experiment is called the sample space of the random experiment, and is denoted by the letter S. s Sample space Important: It s the question that determines the sample space. A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)? S = {HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM } Note: 8 elements, 2 3 Toss a coin once: S = {H, T}. Toss a coin twice: S = {HH, HT, TH, TT} Roll a dice: S = {1, 2, 3, 4, 5, 6} Chose a person at random and check his/her blood type: S = {A,B,AB,O}. A basketball player shoots three free throws. What is the number of baskets made? S = {0, 1, 2, 3} An Event An event is an outcome or collection of outcomes of a random experiment. Events are denoted by capital letters (other than S, which is reserved for the sample space). : tossing a coin 3 times. The sample space in this case is: S = {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT} We can define the following events: Event A: "Getting no H" Event B: "Getting exactly one H" Event C: "Getting at least one H" Event A: "Getting no H" --> TTT Event B: "Getting exactly one H" --> HTT, THT, TTH Event C: "Getting at least one H" --> HTT, THT, TTH, THH, HTH, HHT, HHH Probability Probability of an Event Once we define an event, we can talk about the probability of the event happening and we use the notation: P(A) - the probability that event A occurs, P(B) - the probability that event B occurs, etc. 0 The event is more likely to 1/2 The event is more likely to 1 NOT occur than to occur occur than to occur The probability of an event tells us how likely is it for the event to occur. The event will NEVER occur The event is as likely to occur as it is NOT to occur The event will occur for CERTAIN 2

3 Equally Likely Outcomes : roll a die If you have a list of all possible outcomes and all outcomes are equally likely, then the probability of a specific outcome is Possible outcomes: S={1,2,3,4,5,6} Each of these are equally likely. Event A: rolling a 2 The probability of rolling a 2 is P(A)=1/6 Event B: rolling a 5 The probability of rolling a 5 is P(A)=1/6 : roll a die Event E: getting an even number. : roll two dice What is the probability of the outcomes summing to five? This is S: {(1,1), (1,2), (1,3), etc.} Since 3 out of the 6 equally likely outcomes make up the event E (the outcomes {2, 4, 6}), the probability of event E is simply P(E)= 3/6 = 1/2. There are 36 possible outcomes in S, all equally likely (given fair dice). Thus, the probability of any one of them is 1/36. P(sum is 5) = P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 * 1/36 = 1/9 = A couple wants three children. What are the arrangements of boys (B) and girls (G)? Genetics tells us that the probability that a baby is a boy or a girl is the same, 0.5. Sample space: {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG} All eight outcomes in the sample space are equally likely. The probability of each is thus 1/8. A couple wants three children. What are the numbers of girls (X) they could have? The same genetic laws apply. We can use the probabilities above to calculate the probability for each possible number of girls. Sample space {0, 1, 2, 3} P(X = 0) = P(BBB) = 1/8 P(X = 1) = P(BBG or BGB or GBB) = P(BBG) + P(BGB) + P(GBB) = 3/8 3

4 1. The probability P(A) for any event A is 0 P(A) If S is the sample space in a probability model, then P(S)=1. 3. For any event A, P(A does not occur) = 1- P(A). s Rule 1: For any event A, 0 P(A) 1 Determine which of the following numbers could represent the probability of an event? % 2/3 s Rule 2: P(sample space) = 1 Rule 3: P(A) = 1 P(not A) It can be written as P(not A) = 1 P(A) or P(A)+P(not A) = 1 What is probability that a randomly selected person does NOT have blood type A? P(not A) = 1 P(A) = = 0.58 Note Rule 3: P(A) = 1 P(not A) It can be written as P(not A) = 1 P(A) or P(A)+P(not A) = 1 In some cases, when finding P(A) directly is very complicated, it might be much easier to find P(not A) and then just subtract it from 1 to get the desired P(A). Odds Odds against event A: P( not A) = expressed as a:b Odds in favor event A: = P( not A) 4

5 Event A: rain tomorrow. The probability of rain tomorrow is 80% P(A) = 0.8 What are the odds against the rain tomorrow? P( not A) = = = = 14 : What are the odds in favor of rain tomorrow? = = = = 41 : P( not A) : lottery The odds in favor of winning a lottery is 1:1250 This means that the probability of winning is 1 = = 0. 08% 1251 Rule 4 We are now moving to rule 4 which deals with another situation of frequent interest, finding P(A or B), the probability of one event or another occurring. In probability "OR" means either one or the other or both, and so, P(A or B) = P(event A occurs or event B occurs or both occur) s Consider the following two events: A - a randomly chosen person has blood type A, and B - a randomly chosen person has blood type B. Since a person can only have one type of blood flowing through his or her veins, it is impossible for the events A and B to occur together. On the other hand...consider the following two events: A - a randomly chosen person has blood type A B - a randomly chosen person is a woman. In this case, it is possible for events A and B to occur together. Disjoint or Mutually Exclusive Events Decide if the Events are Disjoint Definition: Two events that cannot occur at the same time are called disjoint or mutually exclusive. Event A: Randomly select a female worker. Event B: Randomly select a worker with a college degree. Event A: Randomly select a male worker. Event B: Randomly select a worker employed part time. Event A: Randomly select a person between 18 and 24 years old. Event B: Randomly select a person between 25 and 34 years old. 5

6 More Rules Rule 4: P(A or B) = P(A) + P(B) Recall: the Addition Rule for disjoint events is P(A or B) = P(A) + P(B) What is the probability that a randomly selected person has either blood type A or B? Since blood type A is disjoint of blood type B, P(A or B) = P(A) + P(B) = = 0.52 But what rule can we use for NOT disjoint events? The general addition rule General addition rule for any two events A and B: The probability that A occurs, or B occurs, or both events occur is: P(A or B) = P(A) + P(B) P(A and B) The general addition rule: example What is the probability of randomly drawing either an ace or a heart from a pack of 52 playing cards? There are 4 aces in the pack and 13 hearts. However, one card is both an ace and a heart. Thus: P(ace or heart) = P(ace) + P(heart) P(ace and heart) = 4/ /52-1/52 = 16/ Note Addition Rules: Summary The General Addition Rule works ALL the time, for ANY two events P(A or B) = P(A) + P(B) P(A and B) Note that if A and B are disjoint events, P(A and B)=0, thus 0 P(A or B) = P(A) + P(B) P(A and B) = P(A) + P(B) Which is the Addition Rule for Disjoint Events. P(A or B)=P(A)+P(B) since P(A and B)=0 P(A or B)=P(A)+P(B)-P(A and B) 6

7 1. The probability P(A) for any event A is 0 P(A) If S is the sample space in a probability model, then P(S)=1. 3. For any event A, P(A does not occur) = 1- P(A). 4. If A and B are disjoint events, P(A or B)=P(A)+P(B). 5. For any two events, P(A or B) = P(A)+P(B)-P(A and B). Probability definition A correct interpretation of the statement The probability that a child delivered in a certain hospital is a girl is 0.50 would be which one of the following? a) Over a long period of time, there will be equal proportions of boys and girls born at that hospital. b) In the next two births at that hospital, there will be exactly one boy and one girl. c) To make sure that a couple has two girls and two boys at that hospital, they only need to have four children. d) A computer simulation of 100 births for that hospital would produce exactly 50 girls and 50 boys. Probability From a computer simulation of rolling a fair die ten times, the following data were collected on the showing face: What is a correct conclusion to make about the next ten rolls of the same die? a) The probability of rolling a 5 is greater than the probability of rolling anything else. b) Each face has exactly the same probability of being rolled. c) We will see exactly three faces showing a 1 since it is what we saw in the first experiment. d) The probability of rolling a 4 is 0, and therefore we will not roll it in the next ten rolls. Probability models If a couple has three children, let X represent the number of girls. Does the table below show a correct probability model for X? a) No, because there are other values that X could be. b) No, because it is not possible for X to be equal to 0. c) Yes, because all combinations of children are represented. d) Yes, because all probabilities are between 0 and 1 and they sum to 1. Probability If a couple has three children, let X represent the number of girls. What is the probability that the couple does NOT have girls for all three children? Probability If a couple has three children, let X represent the number of girls. What is the probability that the couple has either one or two boys? a) b) = c) = a) b) = c) = d)

8 More rules: P(A and B) Another situation of frequent interest, finding P(A and B), the probability that both events A and B occur. P(A and B)= P(event A occurs and event B occurs) As for the Addition rule, we have two versions for P(A and B). Independent events Two events are independent if the probability that one event occurs on any given trial of an experiment is not influenced in any way by the occurrence of the other event. : toss a coin twice Event A: first toss is a head (H) Event B: second toss is a tail (T) Events A and B are independent. The outcome of the first toss cannot influence the outcome of the second toss. Imagine coins spread out so that half were heads up, and half were tails up. Pick a coin at random. The probability that it is headsup is 0.5. But, if you don t put it back, the probability of picking up another heads-up coin is now less than 0.5. Without replacement, successive trials are not independent. In this example, the trials are independent only when you put the coin back ( sampling with replacement ) each time. A woman's pocket contains 2 quarters and 2 nickels. She randomly extracts one of the coins and, after looking at it, replaces it before picking a second coin. Let Q1 be the event that the first coin is a quarter and Q2 be the event that the second coin is a quarter. Are Q1 and Q2 independent events? YES! Why? Since the first coin that was selected is replaced, whether Q1 occurred (i.e., whether the first coin was a quarter) has no effect on the probability that the second coin is a quarter, P(Q2). In either case (whether Q1 occurred or not), when we come to select the second coin, we have in our pocket: More s EXAMPLE: Two people are selected at random from all living humans. Let B1 be the event that the first person has blue eyes and B2 be the event that the second person has blue eyes. In this case, since the two were chosen at random, whether the first person has blue eyes has no effect on the likelihood of choosing another blue eyed person, and therefore B1 and B2 are independent. On the other hand... EXAMPLE: A family has 4 children, two of whom are selected at random. Let B1 be the event that the first child has blue eyes, and B2 be the event that the second chosen child has blue eyes. In this case B1 and B2 are not independent since we know that eye color is hereditary, so whether the first child is blue-eyed will increase or decrease the chances that the second child has blue eyes, respectively. Decide whether the events are independent or dependent Event A: a salmon swims successfully through a dam Event B: another salmon swims successfully through the same dam Event A: parking beside a fire hydrant on Tuesday Event B: getting a parking ticket on the same Tuesday 8

9 The Multiplication Rule for independent events If A and B are independent, P(A and B) = P(A) P(B) The Multiplication Rule for independent events: Roll two dice. Event A: roll a 6 on a red die Event B: roll a 5 on a blue die. What is the probability that if you roll both at the same time, you will roll a 6 on the red die and a 5 on the blue die? Since the two dice are independent, P(A and B)=P(A) P(B)=1/6 1/6 = 1/36 Recall the blood type example: Two people are selected at random from all living humans. What is the probability that both have blood type O? Let O1= "the first has blood type O" and O2= "the second has blood type O" We need to find P(O1 and O2) Since the two were chosen at random, the blood type of one has no effect on the blood type of the other. Therefore, O1 and O2 are independent and we may apply Rule 5: P(O1 and O2) = P(O1)*P(O2) =.44*.44= The probability P(A) for any event A is 0 P(A) If S is the sample space in a probability model, then P(S)=1. 3. For any event A, P(A does not occur) = 1- P(A). 4. If A and B are disjoint events, P(A or B)=P(A)+P(B). 5. General Addition Rule: For any two events, P(A or B) = P(A)+P(B)-P(A and B). 6. If A and B are independent, P(A and B) = P(A) P(B) One more rule We need a general rule for P(A and B). For this, first we need to learn Conditional probability. Notation: P(B A). This means the probability of event B, given that event A has occurred. Event A represents the information that is given. All the students in a certain high school were surveyed, then classified according to gender and whether they had either of their ears pierced: 9

10 Suppose a student is selected at random from the school. Let M and not M denote the events of being male and female, respectively, and E and not E denote the events of having ears pierced or not, respectively. M not M E not E What is the probability that the student has one or both ears pierced? Since a student is chosen at random from the group of 500 students out of which 324 are pierced, P(E)=324/500=.648 What is the probability that the student is a male? Since a student is chosen at random from the group of 500 students out of which 180 are males, P(M)=180/500=.36 What is the probability that the student is male and has ear(s) pierced? Since a student is chosen at random from the group of 500 students out of which 36 are males and have their ear(s) pierced, P(M and E)=36/500=.072 Conditional Probability Now something new: Given that the student that was chosen is a male, what is the probability that he has one or both ears pierced? We will write "the probability of having one or both ears pierced (E), given that a student is male (M)" as P(E M). We call this probability the conditional probability of having one or both ears pierced, given that a student is male: it assesses the probability of having pierced ears under the condition of being male. The total number of possible outcomes is no longer 500, but has changed to 180. Out of those 180 males, 36 have ear(s) pierced, and thus: P(E M)=36/180=0.20. General formula P( Aand B) P( B A) = The above formula holds as long as P(A)>0 since we cannot divide by 0. In other words, we should not seek the probability of an event given that an impossible event has occurred. On the "Information for the Patient" label of a certain antidepressant it is claimed that based on some clinical trials, there is a 14% chance of experiencing sleeping problems known as insomnia (denote this event by I), there is a 26% chance of experiencing headache (denote this event by H), and there is a 5% chance of experiencing both side effects (I and H). Thus, P(I)=0.14 P(H)=0.26 P(I and H)=

11 P(I)=0.14 P(H)=0.26 P(I and H)=0.05 Important!!! (a) Suppose that the patient experiences insomnia; what is the probability that the patient will also experience headache? Since we know (or it is given) that the patient experienced insomnia, we are looking for P(H I). According to the definition of conditional probability: P(H I)=P(H and I)/P(I)=.05/.14=.357. (b) Suppose the drug induces headache in a patient; what is the probability that it also induces insomnia? Here, we are given that the patient experienced headache, so we are looking for P(I H). P(I H)=P(I and H)/P(H)=.05/.26= In general, P( A B) P( B A) Independence Recall: two events A and B are independent if one event occurring does not affect the probability that the other event occurs. Now that we've introduced conditional probability, we can formalize the definition of independence of events and develop four simple ways to check whether two events are independent or not. Independence Checking This example illustrates that one method for checking whether two events are independent is to compare P(B A) and P(B). If P( B A) = P( B) then the two events, A and B, are independent. If P( B A) P( B) then the two events, A and B, are not independent (they are dependent). Similarly, using the same reasoning, we can compare P(A B) and P(A). Recall the side effects example. there is a 14% chance of experiencing sleeping problems known as insomnia (I), there is a 26% chance of experiencing headache (H), and there is a 5% chance of experiencing both side effects (I and H). Thus, P(I)=0.14 P(H)=0.26 P(I and H)=0.05 Are the two side effects independent of each other? To check whether the two side effects are independent, let's compare P(H I) and P(H). In the previous part of this lecture, we found that P(H I)=P(H and I)/P(I)=.05/.14=.357 while P(H)=.26 P(H I) P(H) Knowing that a patient experienced insomnia increases the likelihood that he/she will also experience headache from.26 to.357. The conclusion, therefore is that the two side effects are not independent, they are dependent. 11

12 General Multiplication Rule Comments P(A and B)=P(A)*P(B A) For independent events A and B, we had the rule P(A and B)=P(A)*P(B). Now, for events A and B that may be dependent, to find the probability of both, we multiply the probability of A by the conditional probability of B, taking into account that A has occurred. Thus, our general multiplication rule is stated as follows: Rule 7: The General Multiplication Rule: For any two events A and B, P(A and B)=P(A)*P(B A) This rule is general in the sense that if A and B happen to be independent, then P(B A)=P(B) is, and we're back to Rule 5 - the Multiplication Rule for Independent Events: P(A and B)=P(A)*P(B). Recall the definition of conditional probability: P(B A)=P(A and B)/P(A). Let's isolate P(A and B) by multiplying both sides of the equation by P(A), and we get: P(A and B)=P(A)*P(B A). That's it...this is the General Multiplication Rule. 1. The probability P(A) for any event A is 0 P(A) If S is the sample space in a probability model, then P(S)=1. 3. For any event A, P(A does not occur) = 1- P(A). 4. If A and B are disjoint events, P(A or B)=P(A)+P(B). 5. General Addition Rule: For any two events, P(A or B) = P(A)+P(B)-P(A and B). 6. If A and B are independent, P(A and B) = P(A) P(B) 7. General Multiplication Rule: P( A and B) = P(A) P(B A) Three students work independently on a homework problem. The probability that the first student solves the problem is The probability that the second student solves the problem is The probability that the third student solves the problem is What is the probability that all are able to solve the problem? a) b) (0.95) (0.85) (0.80) c) d) 1 - (0.95) (0.85) (0.80) e) 0.80 Three students work independently on a homework problem. The probability that the first student solves the problem is The probability that the second student solves the problem is The probability that the third student solves the problem is What is the probability that the first student solves the problem and the other two students do not? a) b) (0.95) (0.15) (0.20) c) d) e) 0.95 Three students work independently on a homework problem. The probability that the first student solves the problem is The probability that the second student solves the problem is The probability that the third student solves the problem is What is the probability that none of the three students solves the problem? a) b) c) 1 (0.95) (0.85) (0.80) d) (0.05) (0.15) (0.20) 12

13 Three students work independently on a homework problem. The probability that the first student solves the problem is The probability that the second student solves the problem is The probability that the third student solves the problem is What is the probability that the first student solves the problem or the second student solves the problem? a) b) (0.95) (0.85) c) d) (0.95) (0.85) (0.20) Three students work independently on a homework problem. The probability that the first student solves the problem is The probability that the second student solves the problem is The probability that the third student solves the problem is What is the probability that at least one of them solves the problem correctly? a) b) (0.95) (0.85) (0.80) c) 1 (0.05) (0.15) (0.20) d) 0.95 e) 0.80 Independence Chris is taking two classes this semester, English and American History. The probability that he passes English is The probability that he passes American History is The probability that he passes both English and American History is Are passing English and passing American History independent events? a) Yes, because the classes are taught by different teachers. b) Yes, because the classes use different skills. c) No, because (0.50) (0.40) d) No, because (0.50) (0.40) (0.60). Chris is taking two classes this semester, English and American History. The probability that he passes English is The probability that he passes American History is The probability that he passes both English and American History is What is the probability Chris passes either English or American History? a) b) c) d) e) (0.50) (0.40) 0.60 Conditional probability At a certain university, 47.0% of the students are female. Also, 8.5% of the students are married females. If a student is selected at random, what is the probability that the student is married, given that the student was female? a) / 0.47 b) 0.47 / c) (0.085) (0.47) d) e) 0.47 Conditional probability Within the United States, approximately 11.25% of the population is left-handed. Of the males, 12.6% are left-handed, compared to only 9.9% of the females. Assume the probability of selecting a male is the same as selecting a female. If a person is selected at random, what is the probability that the selected person is a left-handed male? a) (0.126) (0.50) b) (0.126) c) (0.1125) (0.50) d) /

### Section 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

### Probability: 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

### I. 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

### Review for Test 2. Chapters 4, 5 and 6

Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair six-sided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than

### Grade 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

### + 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

### Probability: The Study of Randomness Randomness and Probability Models

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4.1 and 4.2 2009 W.H. Freeman and Company Objectives (IPS Chapters 4.1 and 4.2) Randomness and Probability models Probability

### V. 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

### Lesson 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.

### Distributions. 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

### Construct and Interpret Binomial Distributions

CH 6.2 Distribution.notebook A random variable is a variable whose values are determined by the outcome of the experiment. 1 CH 6.2 Distribution.notebook A probability distribution is a function which

### 33 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

### Basic 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

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

### Probability. 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

### AP 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

### Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.

Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive

### PROBABILITY. 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

### Lecture 2: Probability

Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability

### Tree Diagrams. on time. The man. by subway. From above tree diagram, we can get

1 Tree Diagrams Example: A man takes either a bus or the subway to work with probabilities 0.3 and 0.7, respectively. When he takes the bus, he is late 30% of the days. When he takes the subway, he is

### MAT 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

### Chapter 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

### Chapter 4 - Practice Problems 1

Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula

### Basic 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.

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### MATH 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

### Basic 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

### What 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

### MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

### 36 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

### Chapter 13 & 14 - Probability PART

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 Why Should We Learn Probability Theory? Dr. Joseph

### Bayesian 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

### Topic 6: Conditional Probability and Independence

Topic 6: September 15-20, 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

### Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab

Lecture 2 Probability BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability that the result

### 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

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

### Counting principle, permutations, combinations, probabilities

Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing

### Worked 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

### Events. 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,

### 5_2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

5_2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A prix fixed menu offers a choice of 2 appetizers, 4 main

### Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate

### **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

### Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

### 1. 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

### 15 Chances, Probabilities, and Odds

15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces

### PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

### Chapter 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,

### Question: What is the probability that a five-card 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

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.

12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.

### Section 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

### 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

### I. 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

### Chapter 14 From Randomness to Probability

Chapter 14 From Randomness to Probability 199 Chapter 14 From Randomness to Probability 1. Sample spaces. a) S = { HH, HT, TH, TT} All of the outcomes are equally likely to occur. b) S = { 0, 1, 2, 3}

### PROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE

PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

### Chapter 1 Axioms of Probability. Wen-Guey Tzeng Computer Science Department National Chiao University

Chapter 1 Axioms of Probability Wen-Guey Tzeng Computer Science Department National Chiao University Introduction Luca Paccioli(1445-1514), Studies of chances of events Niccolo Tartaglia(1499-1557) Girolamo

### Probability distributions

Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

### Basics 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.

### Probability and Counting

Probability and Counting Basic Counting Principles Permutations and Combinations Sample Spaces, Events, Probability Union, Intersection, Complements; Odds Conditional Probability, Independence Bayes Formula

### Laws of probability. Information sheet. Mutually exclusive events

Laws of probability In this activity you will use the laws of probability to solve problems involving mutually exclusive and independent events. You will also use probability tree diagrams to help you

### Probability 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

### PROBABILITY 14.3. section. The Probability of an Event

4.3 Probability (4-3) 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

Chance deals with the concepts of randomness and the use of probability as a measure of how likely it is that particular events will occur. A National Statement on Mathematics for Australian Schools, 1991

### Probability. 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

### Probability Lesson #2

Probability Lesson #2 Sample Space A sample space is the set of all possible outcomes of an experiment. There are a variety of ways of representing or illustrating sample spaces. Listing Outcomes List

### Mathematical goals. Starting points. Materials required. Time needed

Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

### The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) State whether the variable is discrete or continuous.

### The Central Limit Theorem Part 1

The Central Limit Theorem Part. Introduction: Let s pose the following question. Imagine you were to flip 400 coins. To each coin flip assign if the outcome is heads and 0 if the outcome is tails. Question:

### Probability 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.

### H + 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

### Introduction to the Practice of Statistics Sixth Edition Moore, McCabe

Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers 5.9 What is wrong? Explain what is wrong in each of the following scenarios. (a) If you toss a fair coin

### 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.

Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are

### Probability and Hypothesis Testing

B. Weaver (3-Oct-25) Probability & Hypothesis Testing. PROBABILITY AND INFERENCE Probability and Hypothesis Testing The area of descriptive statistics is concerned with meaningful and efficient ways of

### Example: Use the Counting Principle to find the number of possible outcomes of these two experiments done in this specific order:

Section 4.3: Tree Diagrams and the Counting Principle It is often necessary to know the total number of outcomes in a probability experiment. The Counting Principle is a formula that allows us to determine

### Chapter 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

### Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

### MTH 110 Chapter 6 Practice Test Problems

MTH 0 Chapter 6 Practice Test Problems Name ) Probability A) assigns realistic numbers to random events. is the branch of mathematics that studies long-term patterns of random events by repeated observations.

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Math 1342 (Elementary Statistics) Test 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the indicated probability. 1) If you flip a coin

### Section 6-5 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)

### PROBABILITY 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

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### Chapter 3. Probability

Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.

### UNIT 7A 118 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS

11 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS UNIT 7A TIME OUT TO THINK Pg. 17. Birth orders of BBG, BGB, and GBB are the outcomes that produce the event of two boys in a family of. We can represent

### MCA SEMESTER - II PROBABILITY & STATISTICS

MCA SEMESTER - II PROBABILITY & STATISTICS mca-5 230 PROBABILITY 1 INTRODUCTION TO PROBABILITY Managers need to cope with uncertainty in many decision making situations. For example, you as a manager may

### Chapter 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

### Sample 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.

### Probability. 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

### CHAPTER 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

### Example: 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

### Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com

Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..

### Probability. 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

### Introduction to Probability

3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which

### Understanding. Probability and Long-Term Expectations. Chapter 16. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Understanding Chapter 16 Probability and Long-Term Expectations Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Thought Question 1: Two very different queries about probability: a. If

### Most of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.

PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe

### Probability 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

### Chapter 4 - Practice Problems 2

Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the