Jim has three cars of different models: A, B, and C. The

If 5% of men and 0.25% of women are color blind, what is the probability that a randomly selected person is color blind?

Suppose that 40% of the students of a campus are women. If 20% of the women and 16% of the men of this campus are A students, what percent of all of them are A students?

Jim has three cars of different models: A, B, and C. The probabilities that models A, B, and C use over 3 gallons of gasoline from Jims house to his work are 0.25, 0.32, and 0.53, respectively. On a certain day, all three of Jims cars have 3 gallons of gasoline each. Jim chooses one of his cars at random, and without paying attention to the amount of gasoline in the car drives it toward his office. What is the probability that he makes it to the office?

One of the cards of an ordinary deck of 52 cards is lost. What is the probability that a random card drawn from this deck is a spade?

Two cards from an ordinary deck of 52 cards are missing. What is the probability that a random card drawn from this deck is a spade?

Of the patients in a hospital, 20% of those with, and 35% of those without myocardial infarction have had strokes. If 40% of the patients have had myocardial infarction, what percent of the patients have had strokes?

Suppose that 37% of a community are at least 45 years old. If 80% of the time a person who is 45 or older tells the truth, and 65% of the time a person below 45 tells the truth, what is the probability that a randomly selected person answers a question truthfully?

A person has six guns. The probability of hitting a target when these guns are properly aimed and fired is 0.6, 0.5, 0.7, 0.9, 0.7, and 0.8, respectively. What is the probability of hitting a target if a gun is selected at random, properly aimed, and fired?

A factory produces its entire output with three machines. Machines I, II, and III produce 50%, 30%, and 20% of the output, but 4%, 2%, and 4% of their outputs are defective, respectively. What fraction of the total output is defective?

Solve the following problem, from the Ask Marilyn column ofParade Magazine, October 29, 2000. I recently returned from a trip to China, where the government is so concerned about population growth that it has instituted strict laws about family size. In the cities, a couple is permitted to have only one child. In the countryside, where sons traditionally have been valued, if the first child is a son, the couple may have no more children. But if the first child is a daughter, the couple may have another child. Regardless of the sex of the second child, no more are permitted. How will this policy affect the mix of males and females? To pose the question mathematically, what is the probability that a randomly selected child from the countryside is a boy?

Suppose that five coins, of which exactly three are gold, are distributed among five persons, one each, at random, and one by one. Are the chances of getting a gold coin equal for all participants? Why or why not?

In a town, 7/9th of the men and 3/5th of the women are married. In that town, what fraction of the adults are married? Assume that all married adults are the residents of the town.

A child gets lost in the Disneyland at the Epcot Center in Florida. The father of the child believes that the probability of his being lost in the east wing of the center is 0.75 and in the west wing is 0.25. The security department sends an officer to the east and an officer to the west to look for the child. If the probability that a security officer who is looking in the correct wing finds the child is 0.4, find the probability that the child is found.

Suppose that there exist N families on the earth and that the maximum number of children a family has is c. Let j ! 0 j c, /c j=0 j = 1 " be the fraction of families with j children. Find the fraction of all children in the world who are the kth born of their families (k = 1, 2, . . . , c).

Let B be an event of a sample space S with P (B) > 0. For a subset A of S, define Q(A) = P (A | B). By Theorem 3.1 we know that Q is a probability function. For E and F, events of S 4 P (FB) > 0 5 , show that Q(E | F ) = P (E | FB).

Suppose that 40% of the students on a campus, who are married to students on the same campus, are female. Moreover, suppose that 30% of those who are married, but not to students at this campus, are also female. If one-third of the married students on this campus are married to other students on this campus, what is the probability that a randomly selected married student from this campus is a woman? Hint: Let M, C, and F denote the events that the random student is married, is married to a student on the same campus, and is female. For any event A, let Q(A) = P (A | M). Then, by Theorem 3.1, Q satisfies the same axioms that probabilities satisfy. Applying Theorem 3.3 to Q and, using the result of Exercise 15, we obtain P (F | M) = P (F | MC)P (C | M) + P (F | MCc )P (Cc | M).

Suppose that the probability that a new seed planted in a specific farm germinates is equal to the proportion of all planted seeds that germinated in that farm previously. Suppose that the first seed planted in the farm germinated, but the second seed planted did not germinate. For positive integers n and k (k < n), what is the probability that of the first n seeds planted in the farm exactly k germinat

Suppose that 10 good and three dead batteries are mixed up. Jack tests them one by one, at random and without replacement. But before testing the fifth battery he realizes that he does not remember whether the first one tested is good or is dead. All he remembers is that the last three that were tested were all good. What is the probability that the first one is also good?

A box contains 18 tennis balls, of which eight are new. Suppose that three balls are selected randomly, played with, and after play are returned to the box. If another three balls are selected for play a second time, what is the probability that they are all new?

From families with three children, a child is selected at random and found to be a girl. What is the probability that she has an older sister? Assume that in a threechild family all sex distributions are equally probable. Hint: Let G be the event that the randomly selected child is a girl, A be the event that she has an older sister, and O, M, and Y be the events that she is the oldest, the middle, and the youngest child of the family, respectively. For any subset B of the sample space let Q(B) = P (B | G); then apply Theorem 3.3 to Q. (See also Exercises 15 and 16.)

Suppose that three numbers are selected one by one, at random and without replacement from the set of numbers {1, 2, 3, . . . , n}. What is the probability that the third number falls between the first two if the first number is smaller than the second?

Avril has certain standards for selecting her future husband. She has n suitors and knows how to compare any two and rank them. She decides to date one suitor at a time randomly. When she knows a suitor well enough, she can marry or reject him. If she marries the suitor, she can never know the ones not dated. If she rejects the suitor, she will not be able to reconsider him. In this process, Avril will not choose a suitor she ranks lower than at least one of the previous ones dated. Avrils goal is to maximize the probability of selecting the best suitor. To achieve this, she adopts the following strategy: For some m, 0 m

(Shrewd Prisoners Dilemma) Because of a prisoners constant supplication, the king grants him this favor: He is given 2N balls, which differ from each other only in that half of them are green and half are red. The king instructs the prisoner to divide the balls between two identical urns. One of the urns will then be selected at random, and the prisoner will be asked to choose a ball at random from the urn chosen. If the ball turns out to be green, the prisoner will be freed. How should he distribute the balls in the urn to maximize his chances of freedom? Hint: Let g be the number of green balls and r be the number of red balls in the first urn. The corresponding numbers in the second urn are N g and N r. The probability that a green ball is drawn is f (g, r): f (g, r) = 1 2 * g g + r + N g 2N g r , . Find the maximum of this function of two variables (r and g). Note that the maximum need not occur at an interior point of the domain.