Lesson 3 Chapter 2: Introduction to Probability

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lesson 3 Chapter 2: Introduction to Probability"

Transcription

1 Lesson 3 Chapter 2: Introduction to Probability Department of Statistics The Pennsylvania State University

2 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of Total Probability and Bayes Theorem 5

3 Overview This chapter presents the basic ideas and tools of classical probability, which is the branch of probability that arose from games of chance. This includes an introduction to combinatorial theory, and an introduction to the concepts of conditional probability and independence. Probability evolved to deal with modeling the randomness of phenomena such as the number of earthquakes, the amount of rainfall, the life time of a given electrical component, or the relation between education level and income, etc. Such probability models will be discussed in Chapters 3 and 4.

4 Sample Spaces Definition The set of all possible outcomes of a random experiment is called the sample space of the experiment, and will be denoted by S. Example a) Give the sample space of the experiment which selects two fuses and classifies each as non-defective or defective. b) Give the sample space of the experiment which selects two fuses and records how many are defective. c) Give the sample space of the experiment which records the number of fuses inspected until the second defective is found.

5 Example Three undergraduate students from a particular university are selected and their opinions about a proposal to expand the use of solar energy are recorded on a scale from 1 to 10. a) Give the sample space of this experiment. What is the size of this sample space? b) Describe the sample space if only the average of the three responses is recorded. What is the size of this sample space?

6 Solution: a) When the opinions of three students are recorded the set of all possible outcomes consist of the triplets (x 1, x 2, x 3 ), where x 1 = 1, 2,..., 10 denotes the response of the first student, x 2 = 1, 2,..., 10 denotes the response of the second student, and x 3 = 1, 2,..., 10 denotes the response of the third student. Thus, the sample space is described as S 1 = {(x 1, x 2, x 3 ) : x 1 = 1, 2,..., 10, x 2 = 1, 2,..., 10, x 3 = 1, 2,..., 10}. There are = 1000 possible outcomes.

7 b) The easiest way to describe the sample space, S 2, when the three responses are averaged is to say that it is the collection of distinct averages (x 1 + x 2 + x 3 )/3 formed from the 1000 triplets of S 1. The word distinct is emphasized because the sample space lists each individual outcome only once, whereas several triplets might result in the same average. For example, the triplets (5, 6, 7) and (4, 6, 8) both yield an average of 6. Determining the size of S 2 is can be done, most easily, with the following R commands: S1=expand.grid(x1=1:10,x2=1:10,x3=1:10) # lists all triplets in S 1 length(table(rowsums(s1))) # gives the number of different sums

8 Events In experiments with many possible outcomes, investigators often classify individual outcomes into distinct categories. For example, the opinion ratings may be classified into low (L = {0, 1, 2, 3}), medium (M = {4, 5, 6}) and high (H = {7, 8, 9, 10}). Definition Such collections of individual outcomes, i.e. subsets of the sample space, are called events. An event consisting of only one outcome is called a simple event. Events are denoted by letters such as A, B, C, etc.

9 Example 1 In selecting one card at random from a deck of cards, the event A = {the card is a spade} consists of 13 outcomes. 2 The event E = {at most 3 heads in five tosses of a coin} consists of the outcomes 0, 1, 2, 3. We say that a particular event A has occurred if the outcome of the experiment is a member of (i.e. contained in) A. The sample space of an experiment is an event which always occurs when the experiment is performed.

10 Set Operations The union, A B, of events A and B, is the event consisting of all outcomes that are in A or in B or in both. The intersection, A B, of A and B, is the event consisting of all outcomes that are in both A and B. The complement, A or A c, of A is the event consisting of all outcomes that are not in A. The events A and B are said to be mutually exclusive or disjoint if they have no outcomes in common. That is, if A B =, where denotes the empty set. The difference A B is defined as A B c. A is a subset of B, A B, if e A implies e B. Two sets are equal, A = B, if A B and B A.

11 Union of A and B A B Intersection of A and B A B A B A B Figure: Venn diagrams for union and intersection

12 The complement of A A c The difference operation A B A A B Figure: Venn diagrams for complement and difference

13 A A B B Figure: Venn diagram illustrations of A, B disjoint, and A B

14 Commutative Laws: a) A B = B A, b) A B = B A Associative Laws: a) (A B) C = A (B C) b) (A B) C = A (B C) Distributive Laws: a) (A B) C = (A C) (B C), b) (A B) C = (A C) (B C) De Morgan s Laws: a) (A B) c = A c B c b) (A B) c = A c B c Two types of proof: (a) By Venn diagram (informal), and (b) Show formally the equality of the two sides, i.e. show that A B and B A, where A is the set on the left and B is the set on the right of each equality.

15 Example The following table classifies a population of 100 plastic disks in terms of their scratch and shock resistance. shock resistance high low scratch high 70 9 resistance low 16 5 Suppose that one of the 100 disks is randomly selected. What is the sample space? Give the outcomes in the events a) the selected disk has low shock resistance, and b) the selected disk has low shock resistance or low scratch resistance.

16 Definition of Probability The Probability Mass Function and Probability Sampling Counting Techniques The probability of an event E, denoted by P(E), is used to quantify the likelihood of occurrence of E by assigning a number from the interval [0, 1]. Higher numbers indicate that the event is more likely to occur. A probability of 1 indicates that the event will occur with certainty, while a probability of 0 indicates that the event will not occur. Read Section

17 Assignment of Probabilities The Probability Mass Function and Probability Sampling Counting Techniques It is simplest to introduce probability in experiments with a finite number of equally likely outcomes, such as those used in games of chance, or simple random sampling. Probability for equally likely outcomes If the sample space consists of N outcomes which are equally likely to occur, then the probability of each outcome is 1/N.

18 Population Proportions as Probabilities The Probability Mass Function and Probability Sampling Counting Techniques A unit is selected by s.r. sampling from a finite statistical population of a categorical variable. If category i has N i units, then the probability the selected unit came from category i is p i = N i /N. where N is the total number of units (so N = N i ). Thus, (a) In rolling a die, the probability of a three is p = 1/6. (b) If 160 out of 500 tin plates have one scratch, and one tin plate is selected at random, the probability that the selected plate has one scratch is p = 160/500.

19 Efron s Dice Outline The Probability Mass Function and Probability Sampling Counting Techniques Die A: four 4s and two 0s Die B: six 3s Die C: four 2s and two 6s Die D: three 5 s and three 1 s Specify the events A > B, B > C, C > D, D > A. Find the probabilities that A > B, B > C, C > D, D > A. Hint: When two dice are rolled, the 36 possible outcomes are equally likely.

20 Outline The Probability Mass Function and Probability Sampling Counting Techniques 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of Total Probability and Bayes Theorem 5

21 The Probability Mass Function and Probability Sampling Counting Techniques Even when population units are selected with equal probability, the outcomes of the random variable recorded may not be equally likely. For example, when die is rolled twice, each of the 36 possible outcomes are equally likely. But if we record the sum of the two rolls, these outcomes are not equally likely. Definition The probability mass function, or pmf, of a discrete random variable X, is a list of the probabilities p(x) for each value x of the sample space S X of X.

22 The Probability Mass Function and Probability Sampling Counting Techniques Example Roll a die twice. Find the pmf of X = the sum of the two die rolls. Solution: S X = {2, 3,..., 12}. This list of possible values, together with the corresponding probabilities, can be found with the R commands: S=expand.grid(X1=1:6,X2=1:6) ; table(s$x1+s$x2) Try also S[which(S$X1+S$X2==7),].

23 The Probability Mass Function and Probability Sampling Counting Techniques It is useful to think of a random experiment as a sampling from the sample space. Example 1 Sampling in the tin plate example can be thought of as sampling from S = {0, 1, 2}. But it should not be simple random sampling (why?). 2 One US citizen aged 18 and over is selected by s.r.sampling and his/her opinion regarding solar energy is rated on the scale 0, 1,..., 10. This can be thought of as random (but not simple random!) sampling from S = {0, 1,..., 10}.

24 The Probability Mass Function and Probability Sampling Counting Techniques Definition When a sample space is thought of as the set from which we sample, we refer to it as sample space population. The random sampling from the sample space population (which is need not be simple random sampling) is called probability sampling, or sampling from a pmf. This idea makes it possible to think of different experiments as sampling from the same population. For example Inspecting 50 products and recording the number of defectives, and Interviewing 50 people and recording if they read New York Times can both be thought as probability sampling from their common sample space S = {0, 1, 2,..., 50}.

25 The Probability Mass Function and Probability Sampling Counting Techniques Example (Simulating an Experiment with R) Use the pmf of X = sum of two die rolls to simulate 1000 repetitions of the experiment which records the sum of two die rolls. Take their mean and use it to guess the population mean. The R commands are S=expand.grid(X1=1:6,X2=1:6) ; pmf=table(s$x1+s$x2)/36 mean(sample(2:12, size=1000, replace=t, prob=pmf))

26 Outline The Probability Mass Function and Probability Sampling Counting Techniques 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of Total Probability and Bayes Theorem 5

27 Why Count? Outline The Probability Mass Function and Probability Sampling Counting Techniques In classical probability counting is used for calculating probabilities. For the probability of an event A we need to know the number of outcomes in A, N(A), and if the sample space consists of a finite number of equally likely outcomes, also the total number of outcomes, N(S), because P(A) = N(A) N(S) Some counting questions are difficult (e.g. how many different five-card hands are possible from a deck of 52 cards?) and thus we need specialized counting techniques.

28 Some Counting Questions The Probability Mass Function and Probability Sampling Counting Techniques 1 How many samples of size n can be formed from N units? The answer is the number of combinations of n objects selected from N, denoted by ( N n), and equals ( ) N N! =, where k! = 1 2 k. n n!(n n)! For example, ( 52 5 ) = 52! = 2, 598, 960 is the number of 5!47! hands of n = 5 cards that can be formed from a deck of N = 52 cards. Knowing that N(S) = 2, 598, 960 we can calculate the probability of individual hands such as the hand with 4 aces and the king of hearts. What is it?

29 The Probability Mass Function and Probability Sampling Counting Techniques 1 What is the probability of A = {the hand has 4 aces}? We need to learn how to determine N(A). 2 When inspecting n items as they come off the assembly line, the probability of the event E = {k of the n inspected items are defective} is calculated using the concept of independence and the answer to the question How many different n-long sequences consisting of k 1s (for defective) and n k 0s (for non defective) can be formed? The answer is (again) the number of combinations ( n k). In what follows we will justify the formula for ( n k). In the process we will learn how to answer question 2.

30 The Product Rules The Probability Mass Function and Probability Sampling Counting Techniques The Simple Product Rule: Suppose a task can be completed in two stages. If stage 1 has n 1 outcomes, and if stage 2 has n 2 outcomes regardless of the outcome in stage 1, then the task has n 1 n 2 outcomes. Example If A = {the hand has 4 aces}, find N(A). Solution. The task is to form a hand with 4 aces. It can be completed in two stages: First select the 4 aces, and then select one additional card. Here n 1 = 1 and n 2 = 48 (why?). Thus, N(A) = 48.

31 The Probability Mass Function and Probability Sampling Counting Techniques Example (1) 1 In how many ways can we select the 1st and 2nd place winners from the four finalists Niki, George, Sophia and Martha? Answer: 4 3 = In how many ways can we select two from Niki, George, Sophia and Martha? Answer: 12 ( ) 4 2 (Why?) Note: 6 = # of combinations =. 2

32 The Probability Mass Function and Probability Sampling Counting Techniques The General Product Rule: If a task can be completed in k stages and stage i has n i outcomes, regardless of the outcomes the previous stages, then the task has n 1 n 2 n k outcomes Example (2) 1 In how many ways can we select a 1st, 2nd and 3rd place winners from Niki, George, Sophia and Martha? Answer: = In how many ways can we select three from Niki, George, Sophia and Martha? Answer: 24 6 (Why?) Note: 4 = # of combinations = ( ) 4. 3

33 Permutations Outline The Probability Mass Function and Probability Sampling Counting Techniques The answer to Example (1), part 1, i.e. 12, is the number of permutations of 2 items selected from 4. The answer to Example (2), part 1), i.e. 24, is the number of permutations of 3 items selected from 4. Definition The number of ordered selections (i.e. when we keep track of the order of selection) of k items from n is called the number of permutations of k items selected from n, it is denoted by P k,n, and equals P k,n = n (n 1)... (n k + 1) = n! (n k)!

34 Combinations The Probability Mass Function and Probability Sampling Counting Techniques In the answer to Example (1), part 2, i.e. 12 2, the 2 in the denominator is the number of permutations of 2 items selected from 2 (P 2,2 = 2 1). In the answer to Example (2), part 2), i.e. 24 6, the 6 in the denominator is the number of permutations of 3 items selected from 3 (P 3,3 = 3 2 1). Extending the rational used to obtain these answers, we have The number of combinations of k items selected from a group of n is (n ) k = P k,n k! = n! k!(n k)!

35 The Probability Mass Function and Probability Sampling Counting Techniques The numbers ( n k) are called binomial coefficients because of the Binomial Theorem: Example (a + b) n = n k=0 ( ) n a k b n k. k a) How many n-long sequences consisting of k 1s and n k zeros can be formed? b) How many paths going from the lower left corner of a 4 3 grid to its upper right corner? Assume one is allowed to move either to the right or upwards.

36 Multinomial Coefficients The Probability Mass Function and Probability Sampling Counting Techniques Suppose we want to assign 8 engineers to work on projects A, B, and C, so that 3 work on project A, 2 work on B, and 3 work on C. In how many ways can this be done? The number of ways n units can be divide in r groups of specified sizes is given by ( ) n = n 1, n 2,..., n r n! n 1!n 2! n r! These numbers are called multinomial coefficients because of the Multinomial Theorem.

37 Example Outline The Probability Mass Function and Probability Sampling Counting Techniques An order comes in for 5 palettes of low grade shingles. In the warehouse there are 10 palettes of high grade, 15 of medium grade, and 20 of low grade shingles. An inexperienced shipping clerk is unaware of the distinction in grades of asphalt shingles and he ships 5 randomly selected palettes. 1 How many different groups of 5 palettes are there? ( 45 5 ) = 1, 221, What is the probability that all of the shipped palettes are low grade? ( 20 5 ) / ( 45 5 ) = 15, 504/1, 221, 759 = What is the probability that 2 of the shipped palettes are of [ medium grade and 3 are from low grade? (15 )( 20 ) ] 2 3 / ( ) 45 5 = ( )/1, 221, 759 = =

38 The Probability Mass Function and Probability Sampling Counting Techniques Example A communication system consists of 15 indistinguishable antennas arranged in a line. The system functions as long as no two non-functioning antennas are next to each other. Suppose six antennas stop functioning. a) How many different arrangements of the six non-functioning antennas result in the system being functional? (Hint: The 9 functioning antennas, lined up among themselves, define 10 possible locations for the 6 non-functioning antennas so the system functions.) b) If the arrangement of the 15 antennas is random, what is the probability the system is functioning?

39 Example Outline The Probability Mass Function and Probability Sampling Counting Techniques What is the probability that 5 randomly dealt cards form a full house? Solution: First, the number of all 5-card hands is ( 52 ) 52! 5 = = 2, 598, 960. Next, think of the task of forming a 5!47! full house as consisting of two stages. In Stage 1 choose two cards of the same kind, and in stage 2 choose three cards of the same kind. Since there are 13 kind of cards, stage 1 can be completed in ( 13 1 )( 4 2) = (13)(6) = 78 ways (why?). For each outcome of stage 1, the task of stage 2 becomes that of selecting three of a kind from one of the remaining 12 kinds. This can be completed in ( 12 1 )( 4 3) = 48 ways. Thus there are (78)(48) = 3, 744 possible full houses, and the desired probability is

40 Reading assignment The Probability Mass Function and Probability Sampling Counting Techniques Read Examples

41 Axioms of Probability The axioms governing any assignment of probabilities are: Axiom 1: P(A) 0, for all events A Axiom 2: P(S) = 1 Axiom 3: If A 1, A 2,... are disjoint P(A 1 A 2...) = P(A i ) i=1

42 Properties of Probability Proposition 1 If A and B are disjoint, P(A B) = 0. 2 If E 1,..., E m are disjoint, then P(E 1 E m ) = P(E 1 ) + + P(E m ) 3 If A B then P(A) P(B). 4 P(A) = 1 P(A c ), for any event A. 5 P(A) = {all simple events E i in A} P(E i)

43 Proposition 1 If S = {s 1,..., s n } and the n outcomes are equally likely, then P(s i ) = 1/n, for all i. 2 P(A B) = P(A) + P(B) P(A B) 3 P(A B C) = P(A) + P(B) + P(C) P(A B) P(A C) P(B C) + P(A B C)

44 Example The probability that a firm will open a branch office in Toronto is 0.7, that it will open one in Mexico City is 0.4, and that it will open an office in at least one of the cities is 0.8. Find the probabilities that the firm will open an office in: 1 neither of the cities 2 both cities 3 exactly one of the cities

45 Example Use the R commands attach(expand.grid(x1=0:1,x2=0:1, X3=0:1,X4=0:1)); table(x1+x2+x3+x4)/length(x1) to find the pmf of the random variable X = number of heads in four flips of a coin. (The answer is x p(x) ) (a) What can we say about the sum of all probabilities? (b) What is P(X 2)? Read also Examples 2.4.2,

46 Updating Probabilities The Law of Total Probability and Bayes Theorem In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the other variable. The updated probabilities are called conditional probabilities. Knowing a man s height helps update the probability that he weighs over 170lb. Knowing a person s education level helps update the probability of that person being in a certain income category.

47 The Law of Total Probability and Bayes Theorem Given partial information regarding the outcome of simple random selection restricts the population. The outcome can be regarded as s.r.s. from the restricted population. Example If the outcome of rolling a die is known to be even, what is the probability it is a 2? If the selected card from a deck is known to be a figure card, what is the probability it is a king? Given event A = {household has a cat}, what is the probability of B = {household has a dog}? http: //stat.psu.edu/ mga/401/fig/venn_square.pdf

48 The Multiplication Rule The Law of Total Probability and Bayes Theorem The conditional probability of the event A given the information that event B has occurred is denoted by P(A B) and equals P(A B) = P(A B), provided P(B) > 0 P(B) THE MULTIPLICATION RULE: The definition of P(A B) yields an alternative formula for P(A B): P(A B) = P(A B)P(B) or P(A B) = P(B A)P(A) The rule extends to more than two events. For example, P(A B C) = P(A)P(B A)P(C A B)

49 The Law of Total Probability and Bayes Theorem Example 1 40% of bean seeds come from supplier A and 60% come from supplier B. Seeds from supplier A have 50% germination rate while those from supplier B have a 75% rate. What is the probability that a randomly selected seed came from supplier A and will germinate? ANSWER: P(A G) = P(G A)P(A) = = Three players are dealt a card in succession. What is the probability that the 1st gets an ace, the 2nd gets a king, and the 3rd gets a queen? ANSWER: P(A B C) = P(A)P(B A)P(C A B) = =

50 The Law of Total Probability and Bayes Theorem Example Fifteen percent of all births involve Cesarean (C) section. Ninety-eight percent of all babies survive delivery (S), whereas, when a C section is performed the baby survives with probability What is the probability that a baby will survive delivery if a C section is not performed? Solution. P(S C) = = (why?) P(S C c ) = = (why?) P(S C c ) = 0.836/0.85 =

51 The Law of Total Probability and Bayes Theorem Example Of the customers entering a department store 30% are men and 70% are women. The probability a male shopper will spend more than $50 is 0.4, and the corresponding probability for a female shopper is 0.6. The probability that at least one of the items purchased is returned is 0.1 for male shoppers and 0.15 for female shoppers. Find the probability that the next customer to enter the department store is a woman who will spend more than $50 on items that will not be returned.

52 Solution. Outline The Law of Total Probability and Bayes Theorem Let W = {customer is a woman}, B = {the customer spends >$50} and R = {at least one of the purchased items is returned}. We want the probability of the intersection of W, B and R c. By the formula for the intersection of three events, this probability is given by P(W B R c ) = P(W )P(B W )P(R c W B) = =

53 The Law of Total Probability and Bayes Theorem The multiplication rule typically applies in situations where the events whose intersection we wish to compute are associated with different stages of an experiment. For example, in the previous example there are three stages: a) record customer s gender, b) record amount spent by customer, and c) record whether any of the items purchased is subsequently returned. Therefore, by the generalized fundamental principle of counting, this experiment has = 8 different outcomes.

54 The Law of Total Probability and Bayes Theorem 0.3 M > 50 < R R c R R c 0.7 W < 50 > R R c R R c Figure: Tree diagram for last example

55 Outline The Law of Total Probability and Bayes Theorem 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of Total Probability and Bayes Theorem 5

56 The Law of Total Probability The Law of Total Probability and Bayes Theorem Let the events A 1, A 2..., A k be disjoint and make up the entire sample space, and let B denote an event whose probability we want to calculate, as in the figure B A A A A If we know P(B A j ) and P(A j ) for all j = 1, 2,..., k, the Law of Total Probability gives P(B) = P(A 1 )P(B A 1 ) + + P(A k )P(B A k )

57 The Law of Total Probability and Bayes Theorem Example 1 40% of bean seeds come from supplier A and 60% come from supplier B. Seeds from supplier A have 50% germination rate while those from supplier B have a 75% rate. What is the probability that a randomly selected seed will germinate? ANSWER: P(G) = P(A)P(G A) + P(B)P(G B) = = Three players are dealt a card in succession. What is the probability that the 2nd gets a king? ANSWER: 4 52 Why?

58 The Law of Total Probability and Bayes Theorem Example Two dice are rolled and the sum of the two outcomes is recorded. What is the probability that 5 happens before 7? Solution 1: If E n = {no 5 or 7 appear on the first n 1 rolls and a 5 appears on the nth}, then P( n=1 E n) = = P(E n ) n=1 ( )n 1 36 = 2 5 n=1

59 The Law of Total Probability and Bayes Theorem Solution 2: Let B be the desired event, A 1 = {first roll results in 5}, A 2 = {first roll results in 7}, A 3 = {first roll results in neither 5 not 7}. Then P(B) = P(B A 1 )P(A 1 ) + P(B A 2 )P(A 2 ) + P(B A 3 )P(A 3 ) = P(A 1 ) P(B)P(A 3 ).

60 The Law of Total Probability and Bayes Theorem Example Two consecutive traffic lights have been synchronized to make a run of green lights more likely. In particular, if a driver finds the first light to be red, the second light will be green with probability 0.9, and if the first light is green the second will be green with probability 0.7. The probability of finding the first light green is 0.6. (a) Find the probability that a driver will find the second traffic light green. (b) Recalculate the probability of part (a) through a tree diagram for the experiment which records whether or not a car stops at each of the two traffic lights.

61 Solution. Outline The Law of Total Probability and Bayes Theorem (a) Let A and B denote the events that a driver will find the first, respectively the second, traffic light green. Because the events A and A c constitute a partition of the sample space, according to the Law of Total Probability P(B) = P(A)P(B A) + P(A c )P(B A c ) = = = (b) The experiment has two outcomes resulting in the second light being green which are represented by the paths with the pairs of probabilities (0.6,0.7) and (0.4,0.9). The sum of the probabilities of these two outcomes is as above.

62 The Law of Total Probability and Bayes Theorem R 0.4 R G 0.6 G R G Figure: Tree diagram for previous example

63 Bayes Theorem The Law of Total Probability and Bayes Theorem Consider events B and A 1,..., A k as in the Law of Total Probability. Now, however, we ask a different question: Given that B has occurred, what is the probability that a particular A j has occurred? The answer is provided by the Bayes theorem: P(A j B) = P(A j B) P(B) = P(A j )P(B A j ) k j=1 P(A i)p(b A i )

64 Bayes Theorem The Law of Total Probability and Bayes Theorem Consider events B and A 1,..., A k as in the Law of Total Probability. Now, however, we ask a different question: Given that B has occurred, what is the probability that a particular A j has occurred? The answer is provided by the Bayes theorem: P(A j B) = P(A j B) P(B) = P(A j )P(B A j ) k j=1 P(A i)p(b A i )

65 Example Outline The Law of Total Probability and Bayes Theorem 1 40% of bean seeds come from supplier A and 60% come from supplier B. Seeds from supplier A have 50% germination rate while those from supplier B have a 75% rate. Given that a randomly selected seed germinated, what is the probability that it came from supplier A? ANSWER: P(A G) = P(A)P(G A) P(A)P(G A) + P(B)P(G B) = Given that the 2nd player got an ace, what is the 3 probability that the 1st got an ace? ANSWER: 51 (Why?)

66 The Law of Total Probability and Bayes Theorem Example Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 10% of the aircraft not discovered have such a locator. Suppose a light aircraft has disappeared. 1 What is the probability that it has an emergency locator and it will not be discovered? 2 What is the probability that it has an emergency locator? 3 If it has an emergency locator, what is the probability that it will not be discovered?

67 Probability of an Intersection The formula for the probability of A B yields P(A B) = P(A) + P(B) P(A B). A simpler formula is possible if A and B are independent: For independent events, P(A B) = P(A)P(B). The above also serves as the definition of independent events.

68 Independent events arise in connection with independent experiments or independent repetitions of the same experiment. Two experiments are independent if there is no mechanism through which the outcome of one experiment will influence the outcome of the other. A die is rolled twice. Are the two rolls independent? Two cards are drawn without replacement from a deck of cards. Are the two draws independent? In two independent repetitions of an experiment, any event associated with the first repetition will be independent of any event associated with the second repetition.

69 Example Toss a coin twice. Find the probability of two heads. Solution. Since the two tosses are independent, P([H in toss 1] [H in toss 2]) = P([H in toss 1])P([H in toss 2]) = = 1 4. Alternatively, since P([H in toss 1] [H in toss 2]) = 1 4 (why?), and also P(H in toss 1)P(H in toss 1) = 1 4, we can conclude that the two events are independent.

70 Example (Fair game with unfair coin) A biased coin results in heads with probability p (e.g. p = 0.3). Flip this coin twice. If the outcome is (H,H) or (T,T) ignore the outcome and flip the coin two more times. Repeat until the outcome of the two flips is either (H,T) or (T,H). In the first case you say you got tails, and in the second case you say you got heads. Prove that now the probability of getting heads equals 0.5. Solution: Ignoring the outcomes (H,H) and (T,T), is equivalent to conditioning on the the event B = {(H, T ), (T, H)}. Thus, P((T, H) B) = P((T, H) B) P(B) = P((T, H)) P(B) = (1 p)p p(1 p) + (1 p)p = 0.5.

71 Example Consider again Efron s dice. That is Die A: four 4s and two 0s; Die B: six 3s; Die C: four 2s and two 6s; Die D: three 5 s and three 1 s. Find the probabilities that A > B, B > C, C > D, and D > A using the properties of probability and the concept of independence. Solution (partial): Note that P(C > D) equals P(C = 2 and D = 1) + P(C = 6 and D = 5 or 1) why? = P(C = 2)P(D = 1) + P(C = 6)P(D = 5 or 1) why? = = 2 3.

72 Independence of Multiple Events When there are several independent experiments, events associated with distinct experiments are independent. But the definition is a bit more complicated: Definition The events A 1,..., A n are mutually independent if P(A i1 A i2... A ik ) = P(A i1 )P(A i2 )... P(A ik ) for any sub-collection A i1,..., A ik of k events chosen from A 1,..., A n All conditions are needed because, for example, P(A B C) = P(A)P(B)P(C) does not imply that A, B, C are independent.

73 Example Consider rolling a die and define the events A = {1, 2, 3}, B = {3, 4, 5}, C = {1, 2, 3, 4}. Verify that P(A B C) = P(A)P(B)P(C), but that A, B are not independent (and thus A, B, C are not mutually independent). Solution: First, since A B C = {3}, it follows that P(A B C) = 1 6 = P(A)P(B)P(C) = Next, A B = {3}, so P(A B) = 1 6 P(A)P(B) =

74 Example The three components of the series system shown in the figure fail with probabilities p 1 = 0.1, p 2 = 0.15 and p 3 = 0.2, respectively, independently of each other. What is the probability the system will fail? Figure: Components connected in series

75 Example The three components of the parallel system shown in figure function with probabilities p 1 = 0.9, p 2 = 0.85 and p 3 = 0.8, respectively, independently of each other. What is the probability the system functions?

76 1 2 3 Figure: Components connected in parallel

77 Read Example

78 Go to previous lesson mga/401/course.info/lesson2.pdf Go to next lesson mga/ 401/course.info/lesson4.pdf Go to the Stat 401 home page http: // mga/401/course.info/

Definition and Calculus of Probability

Definition and Calculus of Probability In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

More information

Lecture 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 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 information

Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

More information

Math/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 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 information

Basic Probability Theory I

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

More information

Probability OPRE 6301

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.

More information

+ Section 6.2 and 6.3

+ 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 information

I. WHAT IS 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

More information

AP Stats - Probability Review

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

More information

ECE302 Spring 2006 HW1 Solutions January 16, 2006 1

ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics

More information

Elements of probability theory

Elements of probability theory 2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

More information

Section 6-5 Sample Spaces and Probability

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)

More information

Basic concepts in probability. Sue Gordon

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

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

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

More information

PROBABILITY 14.3. section. The Probability of an Event

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

More information

Probability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.

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

More information

Basic Probability Concepts

Basic Probability Concepts page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

More information

Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

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

More information

Worked examples Basic Concepts of Probability Theory

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

More information

MAT 1000. Mathematics in Today's World

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

More information

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics

More information

A Few Basics of Probability

A Few Basics of Probability A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

More information

CONTINGENCY (CROSS- TABULATION) TABLES

CONTINGENCY (CROSS- TABULATION) TABLES CONTINGENCY (CROSS- TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods

More information

Statistics 100A Homework 2 Solutions

Statistics 100A Homework 2 Solutions Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

More information

Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

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

More information

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

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.

More information

STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia

STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

Discrete 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 information

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

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

More information

Lecture 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 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 information

E3: PROBABILITY AND STATISTICS lecture notes

E3: 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 information

Chapter 13 & 14 - Probability PART

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

More information

Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical

More information

Statistics in Geophysics: Introduction and Probability Theory

Statistics in Geophysics: Introduction and Probability Theory Statistics in Geophysics: Introduction and Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/32 What is Statistics? Introduction Statistics is the

More information

4.4 Conditional Probability

4.4 Conditional Probability 4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

More information

Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

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 information

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. 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 information

Fundamentals of Probability

Fundamentals of Probability Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

More information

Probability definitions

Probability definitions Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating

More information

Unit 19: Probability Models

Unit 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 information

6.3 Conditional Probability and Independence

6.3 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 information

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

More information

Notes on Probability. Peter J. Cameron

Notes on Probability. Peter J. Cameron Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some

More information

Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014 Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

More information

Conditional Probability and General Multiplication Rule

Conditional Probability and General Multiplication Rule Conditional Probability and General Multiplication Rule Objectives: - Identify Independent and dependent events - Find Probability of independent events - Find Probability of dependent events - Find Conditional

More information

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball. Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw

More information

Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4? Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than

More information

The Calculus of Probability

The Calculus of Probability The Calculus of Probability Let A and B be events in a sample space S. Partition rule: P(A) = P(A B) + P(A B ) Example: Roll a pair of fair dice P(Total of 10) = P(Total of 10 and double) + P(Total of

More information

Introduction to Probability

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

More information

Study Manual for Exam P/Exam 1. Probability

Study Manual for Exam P/Exam 1. Probability Study Manual for Exam P/Exam 1 Probability Eleventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com

More information

Remember to leave your answers as unreduced fractions.

Remember to leave your answers as unreduced fractions. Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,

More information

Ch. 12.1: Permutations

Ch. 12.1: Permutations Ch. 12.1: Permutations The Mathematics of Counting The field of mathematics concerned with counting is properly known as combinatorics. Whenever we ask a question such as how many different ways can we

More information

Jan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 50-54)

Jan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 50-54) Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0- Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample

More information

2 Elementary probability

2 Elementary probability 2 Elementary probability This first chapter devoted to probability theory contains the basic definitions and concepts in this field, without the formalism of measure theory. However, the range of problems

More information

Chapter 5 A Survey of Probability Concepts

Chapter 5 A Survey of Probability Concepts Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible

More information

Basic Probability Theory II

Basic 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 information

Study Manual for Exam P/Exam 1. Probability

Study Manual for Exam P/Exam 1. Probability Study Manual for Exam P/Exam 1 Probability Seventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

Chapter 4 Probability

Chapter 4 Probability 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?

More information

m (t) = e nt m Y ( t) = e nt (pe t + q) n = (pe t e t + qe t ) n = (qe t + p) n

m (t) = e nt m Y ( t) = e nt (pe t + q) n = (pe t e t + qe t ) n = (qe t + p) n 1. For a discrete random variable Y, prove that E[aY + b] = ae[y] + b and V(aY + b) = a 2 V(Y). Solution: E[aY + b] = E[aY] + E[b] = ae[y] + b where each step follows from a theorem on expected value from

More information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 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 information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

More information

Bayesian Tutorial (Sheet Updated 20 March)

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

More information

Probability & Probability Distributions

Probability & Probability Distributions Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions

More information

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

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

More information

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 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 information

Determine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.

Determine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard. Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5

More information

3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

More information

Math 150 Sample Exam #2

Math 150 Sample Exam #2 Problem 1. (16 points) TRUE or FALSE. a. 3 die are rolled, there are 1 possible outcomes. b. If two events are complementary, then they are mutually exclusive events. c. If A and B are two independent

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

More information

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. 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.

More information

Review for Test 2. Chapters 4, 5 and 6

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

More information

The Story. Probability - I. Plan. Example. A Probability Tree. Draw a probability tree.

The Story. Probability - I. Plan. Example. A Probability Tree. Draw a probability tree. Great Theoretical Ideas In Computer Science Victor Adamchi CS 5-5 Carnegie Mellon University Probability - I The Story The theory of probability was originally developed by a French mathematician Blaise

More information

Mathematical goals. Starting points. Materials required. Time needed

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

More information

3. Discrete Probability. CSE 312 Autumn 2011 W.L. Ruzzo

3. Discrete Probability. CSE 312 Autumn 2011 W.L. Ruzzo 3. Discrete Probability CSE 312 Autumn 2011 W.L. Ruzzo sample spaces Sample space: S is the set of all possible outcomes of an experiment (Ω in your text book Greek uppercase omega) Coin flip: S = {Heads,

More information

P (A) = lim P (A) = N(A)/N,

P (A) = lim P (A) = N(A)/N, 1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

More information

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

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.

More information

The study of probability has increased in popularity over the years because of its wide range of practical applications.

The study of probability has increased in popularity over the years because of its wide range of practical applications. 6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

More information

6. Jointly Distributed Random Variables

6. Jointly Distributed Random Variables 6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.

More information

Homework 8 Solutions

Homework 8 Solutions CSE 21 - Winter 2014 Homework Homework 8 Solutions 1 Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that

More information

8.3 Probability Applications of Counting Principles

8.3 Probability Applications of Counting Principles 8. Probability Applications of Counting Principles In this section, we will see how we can apply the counting principles from the previous two sections in solving probability problems. Many of the probability

More information

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 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

More information

Probabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I

Probabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with

More information

ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

More information

Section 6.2 Definition of Probability

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

More information

Stats Review Chapters 5-6

Stats Review Chapters 5-6 Stats Review Chapters 5-6 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant

More information

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

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

More information

The Casino Lab STATION 1: CRAPS

The Casino Lab STATION 1: CRAPS The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

More information

Section 5-3 Binomial Probability Distributions

Section 5-3 Binomial Probability Distributions Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial

More information

Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli

More information

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions... MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

More information

Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

More information

Solutions to Self-Help Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)}

Solutions to Self-Help Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)} 1.4 Basics of Probability 37 Solutions to Self-Help Exercises 1.3 1. Consider the outcomes as ordered pairs, with the number on the bottom of the red one the first number and the number on the bottom of

More information

Chapter 3. Probability

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.

More information

Probability distributions

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,

More information

Topic 1 Probability spaces

Topic 1 Probability spaces CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in five-card poker is about 0.2%,

More information

AP Statistics 7!3! 6!

AP Statistics 7!3! 6! Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

More information

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

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

More information

Probability and Hypothesis Testing

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

More information