CH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?

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1 CH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? P7. The king comes from a family of 2 children. what is the probability that the other child is his sister? P12. A recent college graduate is planning to take the first three acuarial examinations in the coming summer. she will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is.9. If she passes the first exam, then the conditional probability that she passes the second one is.8, and if she passes both the first and the second exams, then the conditonal probability that she passes the third exam is.7. (a) What is the probability that she passes all three exams? (b) Given that she did not pass all three exams, what is the conditional probability that she failed the second exam? P17. In a certain community, 36 percent of the families own a dog, and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is (a) the probability that a randomly selected family owns both a dog and a cat; (b) the conditional probability that a randomly selected family owns a dog given that it owns a cat? P20. Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that (a) the student is female, given that the student is majoring in computer science; (b) the student is majoring in computer science, given that the student is female. 1

2 P52. A high school student is anxiously waiting to receive mail telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities, given that she is accepted and that she is rejected, of receiving notification on each day of next week are as follows: Day P (mail accepted) P (mail rejected) Monday Tudeday Wednesday Thursday Friday She estimates that her probability of being accepted is.6. (a) What is the probability that mail is received on Monday? (b) What is the probability that mail is received on Tudesday given that it is not received on Monday? (c) If there is no mail through Wednesday, what is the conditonal probability that she will be accepted? (d) What is the conditional probability that she will be accepted if mail comes on Thursday? (e) What is the conditional probability that she will be accepted if no mail arrives that week? T1. If P (A) > 0, show that P (AB A) P (AB A B) T2. If A B, express the following probabilities as simply as possible: P (A B), P (A B c ), P (B A), P (B A c ) 2

3 T6. Prove that if E 1, E 2,..., E n are independent events, then n P (E 1 E 2... E n ) = 1 [1 P (E i )] i=1 T9. Consider two inependent tosses of a fair coin. Let A be the event that the first toss lands heads, let B be the event that the second toss lands heads, and let C be the event that both land on the same side. Show that the events A, B, C are pairwise independent that is, A and B are independent, A and C are independent, and B and C are independent but not independent. CH 3 P36. Stores A, B, and C have 50, 75, and 100 employees and, respectively, 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. one employee resigns, and this is a woman. What is the probability that she works in store C? P42. Three cooks, A, B, and C, bake a special kind of cake, and with respective probabilities.02,.03, and.05 it fails to rise. In the restaurant where they work, A bakes 50 percent of these cakes, B 30 percent, and C 20 percent. What proportion of failures is caused by A? P53. A parallel system functions whenever at least one of its components work. Consider a parallel system of n components and suppose that each component independently works with probability 1. Find the conditional probability that component 1 works given that the system is 2 functioning. 3

4 P55. In a class there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random? P64. A true-false question is to be posed to a husband and wife team on a quiz show. Both the husband and the wife will, independently, give the correct answer with probability p. Which of the following is a better strategy for this couple? (a) Choose one of them and let that person answer the question; or (b) have them both consider the question and then either give the common answer if they agree or, if they disagree, flip a coin to determine which answer to give? P66. The probability of the closing of the ith relay in the circuits shown in Figure 3.4 is given by p i, i=1, 2, 3, 4, 5. If all relays function independently, what is the probability that a corrent flows between A and B for the respective circuits? 1 2 (a) A 5 B (b) A 4 3 B 2 5 T28. Prove or given a counterexample. If E 1 and E 2 are independent, then they are conditionally independent given F. 4

5 CH4 P1. Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value? P2. Two fair dice are rolled. Let X equal the product of the 2 dice. Compute P {X = i} for i = 1, 2,... P13. A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability.3, and his second will lead independently to a sale with probability.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. Determine the probability mass function of X, the total dollar value of all sales. CH 4 P19. If the distribution function of X is given by 0 b < b < b < 2 F (b) = b < b < b 3.5 calculate the probability mass function of X. 5

6 P20. A gambling book recommends the following winning strategy for the game of roulette. It recommends that a gambler bet $1 on red. If red appears (which has probability 18 ), then the gambler should take her $1 profit and 38 quit. If the gambler loses this bet(which has probability 20 of occurring), she 38 should make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote the gambler s winnings when she quits. (a) Find P {X > 0}. (b) Are you convinced that the strategy is indeed a winnings stragtegy? Explain you answer! (c) Find E[X]. P33. A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with n=10, p= 1, approximately how many papers 3 should he purchase so as to maximize his expected profit? P38. If E[X]=1 and V ar(x)=5, find (a) E[(2 + X) 2 ]; (b) V ar(4 + 3X). P42. Suppose that when in flight, airplane engines will fail with probability 1 p independently from engine to engine. If an airplane needs a majority of its engines operative to make a successful flight, for what values of p is a 5-engine plane preferable to a 3-engine plane? P51. The expected number of typographical errors on a page of a certain magazine is.2. What is the probability that the next page you read contains (a) 0 and (b) 2 or more typographical errors? Explain your reasoning! 6

7 P61. The probability of being dealt a full house in a hand of poker is approximately Find an approximation for the probability that in 1000 hands of poker you will be dealt at least 2 full houses. P71. Consider a roulette wheel consisting of 38 numbers 1 through36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12, what is the probability that (a) Smith will lose his first 5 bets; (b) his first win will occur on his fourth bet? T6. For a nonnegative integer-valued random variable N, show that E[N] = P {N i} i=1 Hint: P {N i} = P {N = k}. Now interchange the order of summation. i=1 i=1 k=i T18. Let X be a Poisson random variable with parameter λ. What value of λ maximizes P {X = k}, k 0? 7

Math/Stat 394 Homework 5

Math/Stat 394 Homework 5 1. If we select two black balls then X 4. This happens with probability ( 4 2). If we select two white balls then X 2. This happens with ( 14 probability (8 2). If we select two