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theory of probability

Questions and Answers of Theory Of Probability

25. Two cards are randomly selected from a deck of 52 playing cards.(a) What is the probability they constitute a pair (that is, that they are of the same denomination)?(b) What is the conditional
26. A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events E1,E2,E3, and E4 as follows:E1 = {the first pile has exactly 1 ace}, E2 = {the
27. Suppose in Exercise 26 we had defined the events Ei , i = 1, 2, 3, 4, by E1 = {one of the piles contains the ace of spades}, E2 = {the ace of spades and the ace of hearts are in different piles},
28. If the occurrence of B makes A more likely, does the occurrence of A make B more likely?
29. Suppose that P(E) = 0.6. What can you say about P(E|F) when(a) E and F are mutually exclusive?(b) E ⊂ F?(c) F ⊂ E?
13. The dice game craps is played as follows. The player throws two dice, and if the sum is seven or eleven, then she wins. If the sum is two, three, or twelve, then she loses.If the sum is anything
12. Let E and F be mutually exclusive events in the sample space of an experiment.Suppose that the experiment is repeated until either event E or event F occurs.What does the sample space of this new
A family has two children. What is the conditional probability that both are boys given that at least one of them is a boy? Assume that the sample space S is given by S = {(b, b), (b, g), (g, b), (g,
Bev can either take a course in computers or in chemistry. If Bev takes the computer course, then she will receive an A grade with probability 12; if she takes the chemistry course then she will
Suppose an urn contains seven black balls and five white balls. We draw two balls from the urn without replacement. Assuming that each ball in the urn is equally likely to be drawn, what is the
There are r players, with player i initially having ni units, ni > 0, i = 1, . . . , r. At each stage, two of the players are chosen to play a game, with the winner of the game receiving 1 unit from
In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer and 1 − p the probability that she guesses. Assume
1. A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from
2. Repeat Exercise 1 when the second marble is drawn without replacing the first marble.
3. A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times?
4. Let E, F,G be three events. Find expressions for the events that of E, F,G(a) only F occurs,(b) both E and F but not G occur,(c) at least one event occurs,(d) at least two events occur,(e) all
5. An individual uses the following gambling system at Las Vegas. He bets $1 that the roulette wheel will come up red. If he wins, he quits. If he loses then he makes the same bet a second time only
6. Show that E(F ∪ G) = EF ∪ EG.
7. Show that (E ∪ F)c = EcFc.
8. If P(E) = 0.9 and P(F) = 0.8, show that P(EF) 0.7. In general, show that P(EF) P(E) + P(F) − 1 This is known as Bonferroni’s inequality.
9. We say that E ⊂ F if every point in E is also in F. Show that if E ⊂ F, then P(F) = P(E) + P(FEc) P(E)
10. Show thatThis is known as Boole’s inequality.H int: Either use Equation (1.2) and mathematical induction, or else show that ni =1 Ei = ni =1 Fi , where F1 = E1, Fi = Ei i−1 j=1 Ecj , and use
11. If two fair dice are tossed, what is the probability that the sum is i, i = 2, 3, . . . , 12?
30. Bill and George go target shooting together. Both shoot at a target at the same time.Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with
31. What is the conditional probability that the first die is six given that the sum of the dice is seven?
32. Suppose all n men at a party throw their hats in the center of the room. Each man then randomly selects a hat. Show that the probability that none of the n men selects his own hat isNote that as
Suppose that an airplane engine will fail, when in flight, with probability 1−p independently from engine to engine; suppose that the airplane will make a successful flight if at least 50 percent
Suppose that a particular trait of a person (such as eye color or left handedness) is classified on the basis of one pair of genes and suppose that d represents a dominant gene and r a recessive
Suppose that the number of typographical errors on a single page of this book has a Poisson distribution with parameter λ = 1. Calculate the probability that there is at least one error on this page.
If the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ = 3, what is the probability that no accidents occur today?
Consider an experiment that consists of counting the number ofα-particles given off in a one-second interval by one gram of radioactive material.If we know from past experience that, on the average,
Calculate the cumulative distribution function of a random variable uniformly distributed over (α, β).
IfX is uniformly distributed over (0, 10), calculate the probability that (a) X < 3, (b) X > 7, (c) 1 < X < 6.
Find E[X] where X is the outcome when we roll a fair die.
Calculate E[X]when X is a Bernoulli random variable with parameter p.
Calculate E[X]when X is binomially distributed with parameters n and p.
Calculate the expectation of a geometric random variable having parameter p.
Calculate E[X] if X is a Poisson random variable with parameter λ.
Calculate the expectation of a random variable uniformly distributed over (α, β).
Let X be exponentially distributed with parameter λ. Calculate E[X].
Calculate E[X] when X is normally distributed with parameters μ and σ2.
Suppose X has the following probability mass function:Calculate E[X2]. p(0) = 0.2, p(1) = 0.5, p(2) = 0.3
It is known that any item produced by a certain machine will be defective with probability 0.1, independently of any other item. What is the probability that in a sample of three items, at most one
Four fair coins are flipped. If the outcomes are assumed independent, what is the probability that two heads and two tails are obtained?
33. In a class there are four freshman boys, six freshman girls, and six sophomore boys.How many sophomore girls must be present if sex and class are to be independent when a student is selected at
34. Mr. Jones has devised a gambling system for winning at roulette. When he bets, he bets on red, and places a bet only when the ten previous spins of the roulette have landed on a black number. He
35. A fair coin is continually flipped. What is the probability that the first four flips are(a) H, H, H, H?(b) T, H, H, H?(c) What is the probability that the pattern T, H, H, H occurs before the
36. Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box.
37. In Exercise 36, what is the probability that the first box was the one selected given that the marble is white?
38. Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from
39. 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
40. (a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair
41. In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling.(a) What is the probability that this rat is a pure black rat (as
42. There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the three coins is
43. Suppose we have ten coins which are such that if the ith one is flipped then heads will appear with probability i/10, i = 1, 2, . . . , 10. When one of the coins is randomly selected and flipped,
44. Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is
45. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn c additional balls of the same color are put in with it. Now suppose that we
46. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which
47. For a fixed event B, show that the collection P(A|B), defined for all events A, satisfies the three conditions for a probability. Conclude from this that P(A|B) = P(A|BC)P(C|B) +
48. Sixty percent of the families in a certain community own their own car, thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is randomly
Let X be uniformly distributed over (0, 1). Calculate E[X3].
25. Independent trials, resulting in one of the outcomes 1, 2, 3 with respective probabilities p1, p2, p3,3i=1 pi = 1, are performed.(a) LetN denote the number of trials needed until the initial
44. The number of customers entering a store on a given day is Poisson distributed with mean λ = 10. The amount of money spent by a customer is uniformly distributed over (0, 100). Find the mean and
45. An individual traveling on the real line is trying to reach the origin. However, the larger the desired step, the greater is the variance in the result of that step.Specifically, whenever the
46. (a) Show that Cov(X,Y) = Cov(X,E[Y |X])(b) Suppose, that, for constants a andb, E[Y |X] = a + bX Show that b = Cov(X, Y)/Var(X)
47. If E[Y |X] = 1, show that Var(XY) Var(X)
48. Suppose that we want to predict the value of a random variable X by using one of the predictors Y1, . . . , Yn, each of which satisfies E[Yi |X] = X. Show that the predictor Yi that minimizes
49. A and B play a series of games with A winning each game with probability p. The overall winner is the first player to have won two more games than the other.(a) Find the probability that A is the
50. There are three coins in a barrel. These coins, when flipped, will come up heads with respective probabilities 0.3, 0.5, 0.7. A coin is randomly selected from among these three and is then
51. If X is geometric with parameter p, find the probability that X is even.
52. Suppose that X and Y are independent random variables with probability density functions fX and fY. Determine a one-dimensional integral expression for P{X +Y < x}.
53. Suppose X is a Poisson random variable with mean λ. The parameter λ is itself a random variable whose distribution is exponential with mean 1. Show that P{X =n} = (12)n+1.
54. A coin is randomly selected from a group of ten coins, the nth coin having a probability n/10 of coming up heads. The coin is then repeatedly flipped until a head appears. Let N denote the number
55. You are invited to a party. Suppose the times at which invitees are independent uniform (0,1) random variables. Suppose that, aside from yourself, the number of other people who are invited is a
56. Data indicate that the number of traffic accidents in Berkeley on a rainy day is a Poisson random variable with mean 9, whereas on a dry day it is a Poisson random variable with mean 3. Let X
57. The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed over (0, 5). That is, is uniformly distributed over (0, 5), and
58. A collection of n coins is flipped. The outcomes are independent, and the ith coin comes up heads with probability αi , i = 1, . . . , n. Suppose that for some value of j, 1 j n, αj = 12.
59. Suppose each new coupon collected is, independent of the past, a type i coupon with probability pi . A total of n coupons is to be collected. Let Ai be the event that there is at least one type i
43. The density function of a chi-squared random variable having n degrees of freedom can be shown to bewhere (t) is the gamma function defined byIntegration by parts can be employed to show that (t)
42. If Xi , i = 1, . . . , n are independent normal random variables, with Xi having meanμi and variance 1, then the random variableni=1 X2 i is said to be a noncentral chi-squared random
26. You have two opponents with whom you alternate play. Whenever you play A, you win with probability pA; whenever you play B, you win with probability pB,where pB > pA. If your objective is to
27. A coin that comes up heads with probability p is continually flipped until the pattern T, T, H appears. (That is, you stop flipping when the most recent flip lands heads, and the two immediately
28. Polya’s urn model supposes that an urn initially contains r red and b blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the
29. Two players take turns shooting at a target, with each shot by player i hitting the target with probability pi , i = 1, 2. Shooting ends when two consecutive shots hit the target. Let μi denote
30. Let Xi , i 0 be independent and identically distributed random variables with probability mass functionFind E[N], where N = min{n > 0 : Xn = X0}. p(j) = P(X;=j}, j=1,...,m, P(j) = 1 j=1
31. Each element in a sequence of binary data is either 1 with probability p or 0 with probability 1 − p. A maximal subsequence of consecutive values having identical outcomes is called a run. For
32. Independent trials, each resulting in success with probability p, are performed(a) Find the expected number of trials needed for there to have been both at least n successes or at least m
33. If Ri denotes the random amount that is earned in period i, then∞i=1 βi−1Ri , where 0 that is independent of the Ri . Show that the expected total discounted reward is equal to the expected
34. A set of n dice is thrown. All those that land on six are put aside, and the others are again thrown. This is repeated until all the dice have landed on six. Let N denote the number of throws
35. Consider n multinomial trials, where each trial independently results in outcome i with probability pi ,ki=1 pi = 1. With Xi equal to the number of trials that result in outcome i, find E[X1|X2
36. Let p0 = P{X = 0} and suppose that 0 < p0 < 1. Let μ = E[X] and σ2 = Var(X).(a) Find E[X|X = 0].(b) Find Var(X|X = 0).
37. A manuscript is sent to a typing firm consisting of typists A, B, and C. If it is typed by A, then the number of errors made is a Poisson random variable with mean 2.6;if typed by B, then the
38. Let U be a uniform (0, 1) random variable. Suppose that n trials are to be performed and that conditional on U = u these trials will be independent with a common success probability u. Compute
39. A deck of n cards, numbered 1 through n, is randomly shuffled so that all n! possible permutations are equally likely. The cards are then turned over one at a time until card number 1 appears.
40. A prisoner is trapped in a cell containing three doors. The first door leads to a tunnel that returns him to his cell after two days of travel. The second leads to a tunnel that returns him to
41. A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to the right, then it will wander around in the maze for three minutes and will then return to its initial
60. Two players alternate flipping a coin that comes up heads with probability p. The first one to obtain a head is declared the winner. We are interested in the probability that the first player to
61. Suppose in Exercise 29 that the shooting ends when the target has been hit twice.Letmi denote the mean number of shots needed for the first hit when player i shoots first, i = 1, 2. Also, let Pi
62. A,B, and C are evenly matched tennis players. Initially A and B play a set, and the winner then plays C. This continues, with the winner always playing the waiting player, until one of the
81. Let Xi , i1, be independent uniform (0, 1) random variables, and define N by N = min{n: Xn < Xn−1}where X0 = x. Let f (x) = E[N].(a) Derive an integral equation for f (x) by conditioning on

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