MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3

Transcription

1 MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3 1. A four engine plane can fly if at least two engines work. a) If the engines operate independently and each malfunctions with probability q, what is the probability that the plane will fly safely? b) A two engine plane can y if at least one engine works and if an engine malfunctions with probability q, what is the probability that plane will fly safely? c) Which plane is the safest? 2. It has been observed empirically that the deaths per hour due to traffic accidents, occur at a rate of 8 per hour on long holiday weekends in the United States. Assuming these deaths occur independently: compute the probability that: a) A 1-hour period would pass with no deaths? b) 15-min period would pass with no deaths? c) 4 consecutive, nonoverlaping 6-min intervals would pass with no deaths? 3. It has been observed that packages of Hamm s beer are removed from the shelf of a particular supermarket at a rate of 10 per hour during rush periods. a) What is the probability that at least one package is removed during the first 6 min of a rush hour? b) What is the probability that at least one package is removed during each of 3 consecutive, non-overlapping 6 min intervals? 4. An absentminded professor does not remember which of his 12 keys will open his office door. If he tries them at random and with replacement a) On the average, how many keys should he try before unlocking his office? b) What is the probability that he opens his office door on his third try? 5. An incoming lot of the material contains 100 items. A sample of 5 items will be inspected; each item selected will be classified as defective or nondefective. If the sample contains 1 or fewer defectives it will be accepted, otherwise it will be rejected. If the lot contains 15 defectives, what is the probability that it will be rejected? 6. A rat maze consists of a straight corridor, at the end of which is a branch; at the branching point the rat must either turn right or left. Assume 10 rats are placed in the maze, one at a time. a) If each is choosing one of the two branches at random, what is the distribution of the number that turn right? b) What is the probability at least 9 will turn the same way? 7. Sharon and Ann play a series of backgammon games until one of them wins five games. Suppose that the games are independent and the probability that Sharon wins a) Find the probability that the series ends in seven games? b) If the series ends in seven games, what is the probability that Sharon wins? 8. A computer store has purchased 3 computers of a certain type at 500 dollars a piece. It will sell them for 1000 dollars a piece. The manufacturer has agreed to purchase any computers still unsold after a specified period at 200 dollars apiece. Let X denote the number of computers sold, and suppose that p X (0) = 0.1, p X (1) = 0.2, p X (2) = 0.3 and

2 p X (3) = 0.4. Find the expected profit. 9. A student who is trying to write a paper for a course has a choice of two topics, A and B. If topic A is chosen, the student will order 2 books through interlibrary loan, while if topic B is chosen, the student will order 4 books. The student feels that a good paper necessitates receiving and using at least half the books ordered for either topic chosen. a) If the probability that a book ordered through interlibrary loan actually arrives on time is 0.9 and books arrive independently of one another, which 2.topic should the student choose to maximize the probability of writing a good paper? b) What if, the arrival probability is only 0.5 instead of 0.9? 10. In a survey of 15 manufacturing firms, the number of firms that use LIFO is a Binomial random variable X with p = 0.2. a) What is the probability that 5 or fewer firms will be found to use LIFO? Is it unlikely that more than 10 firms will be found to use LIFO? Comment. b) Plot the probability mass funcion of X. Is the distribution highly skewed? c) What are the mean and the standard deviation of X? What is the coefficient of variation of X? What would be the value of this coefficient if 150 firms were surveyed rather than 15 (continue to assume that p = 0.2)? Describe the effect of the larger survey on the relative variability of X. 11. A second stage smog alert has been called in a certain area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations. a) If 15 of the firms are actually violating at least one regulation, what is the probability mass function of the number of firms visited by the inspector that are in violation of at least one regulation? b) If there are 500 firms in the area of which 150 are in violation, find P( 2) by the probability mass function found in part a) and approximate it with a simpler probability mass function. c) Let X be the number among the 10 visited that are in violation, compute E(X) and Var(X) both for the exact probability mass function and the approximating probability mass function in b). 12. Individual A has a red die and B has a green die (both fair). If they each roll until they obtain 5 doubles (1-1, 2-2,...6-6) a) What is the probability mass function of X where X is the total number of times a die is rolled? b) Find EX. c) Find Var(X). 13. Suppose that two teams I and II play a series of at most 7 games with the convention that the first team to win 4 games wins the series. Suppose also that the outcomes of the games are independent of each other and that team I has a constant probability p of winning on each game. a) What is the probability that team I wins for the 4 th time on the k th game and what is the probability that team I wins the series? b) What is the probability that the series will end in exactly 6 games?

3 14. The number of prescription for a certain medication that are written between 9:00-9:30 by a physician is a Poisson random variable with = 2.5. a) Obtain the probability of no prescription during the period and the probability of two or fewer prescriptions. b) What is the expected number of prescriptions written by the physician during this period? What is the variance of the number of prescriptions? c) Plot the probability mass function of X. Is the distribution skewed? If so, in which direction? 15. The probability that a certain baseball player gets a hit is 0.3 and we assume that times at bat are independent. a) What is the probability that he will require 5 times at bat to get his first hit? Which distribution is this? b) What is the probability that he will require 5 more times at bat to get his first hit given that he has been at bat 10 times without a hit? 16. In a clinical trial with two treatment groups, the probability of succeess in one treatment group is 0.5, and the probability of success in the other is 0.6. Suppose that there are 5 patients in each group. Assume that the outcomes of all patients are independent. Calculate the probability that the first group will have at least as many successes as the 2nd group. 17. Suppose that n students are selected at random without replacement from a class containing T students, of whom A are boys and T-A are girls. Let X denote the number of boys that are obtained. For what sample size n will Var(X) be a maximum? 18. An airline sells 200 tickets for a certain flight on an airplane that has only 198 seats because, on the average, 1 percent of purchasers of airline tickets do not appear for the departure of their flight. Determine the probability that everyone who appears for the departure of this flight will have a seat. 19. Suppose that the average number of accidents occuring weekly on a particular stretch of a highway equals 3. Calculate the probability that there is at least 4 accidents this month. (Take one month = 4 weeks) 20. Suppose that a sequence of independent tosses are made with a balanced coin for which the probability of obtaining a head on each given toss is 1/3. Given that no head is observed before 5th toss, what is the probability that no head will be observed before 8th toss? 21. A certain electric system contains 10 components. Suppose that the probability that each individual component will fail is 0.2 and that the components fail independently of each other. Given that at least one of the components has failed, what is the probability that at least two of the components have failed? 22. At a county fair, a ring toss gane may be played for 25 cent. You are given three rings and then attempt to toss them individually onto a peg. If you successfully get one ring on a peg, you win a prize woth 50 cents. If you get two on, you get a prize worth 1 dollar and if you get all three on, you win a prize worth 5 dollars. Assuming the probability that you ring the peg is 0.1 each try, a) Find the probability mass function of your gain if you play this game one time? b) What is your expected gain if you play this game one time?

4 c) What is your expected gain if you play this game ten times? 23. A coin is ipped until the first head occurs. Assume the ips are independent and that the probability of a head occuring each time is p a) Show that the probability that an odd number of ips is required is p/(1-q 2 ) where q = 1-p. b) Find the value of p such that the probability is 0.4 that an odd number of ips is required. 24. Suppose that there are N 2 machines in the system which work simultaneously and independently of each other. The system fails when more than two of the machines in the system fail. If the number of periods until a single machine fails has a geometric distribution with parameter p, a) Find the probability that the system will be in a working state after T periods of time. b) Find the expected number of failed machines at the end of T periods. 25. An incoming lot of material contains N items. A sample of k items will be inspected, which will classify the selected items to be defective or non-defective. If the sample contains one or fewer defectives, the lot will be accepted. Otherwise it will be rejected. a) If the lot contains s defective items, what is the probability that the lot will be accepted? b) If 6 lots, containing s defective items, come in a week, what is the probability that exactly 2 lots will be accepted? What is the expected value and variance of the lots accepted? 26. Suppose X is a random variable with the following moment generating function: 1 M t e e e 8 t t 2t X ( ) (1 2 4 ) a) Find EX. b) Find VarX. c) Find the probability mass and distribution function of X. 27. Suppose the number of customers, arriving at a market is modeled as a Poisson Process with mean arrival rate = 4/hour. a) If exactly 2 customers arrive during 9:00-9:30 what is the probability that at least 8 customers arrive during 9:00-10:30? b) If less than 20 customers arrive during 9:00-17:30, the market is assumed to be nonprofitable. What is the probability that exactly 1 day will be profitable in a week? (Assume that market is open 6 days in a week) 28. Suppose that the random variables X1, X 2,..., X n form n Bernoulli trials with parameter p. Determine the conditional probability that X 1 = 1 given that i n 1 X k( k 1,2,..., n) i 29. In a given semester, a university will process grades. In the past, 0.1% of the grades have been erroneously reported. Assume you are taking 5 courses at this university in one semester. What is the probability that all your grades are correctly reported? 30. Assume 1 baby in is born blind. If a large city hospital had 5000 births in 1970, approximate the probability that none of the babies born that year was blind at birth. Also

5 approximate the probability that exactly one was born blind and that at least two were born blind. 31. A box contains 7 light bulbs, of which 2 are bad. The bulbs are tested one after another without replacement. a) Let X be the number of bulbs tested until the first bad bulb is found. Find P(X = x) and FX(x). b) Let Y be the number of bulbs tested to locate the second bad bulb. What is P(Y = y) and F Y (y).