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Chapter 3: Q.3.31 (page 111)
There is a percent chance that event will occur. If does not occur, then there is a percent chance that will occur. What is the probability that at least one of the events will occur?
The probability that at least one of the events or will occur
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Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than \(250,000 per year, while a total of 3.3 percent of all California households earn more than \)250,000 per year
(a) What proportion of all non-California households earn more than \(250,000 per year?
(b) Given that a randomly chosen U.S. household earns more than \)250,000 per year, what is the probability it is a California household
A total of 500 married working couples were polled about their annual salaries, with the following information resulting:
For instance, in 36 of the couples, the wife earned more and the husband earned less than \( 25,000. If one of the couples is randomly chosen, what is
(a) the probability that the husband earns less than \) 25,000 ?
(b) the conditional probability that the wife earns more than \( 25,000 given that the husband earns more than this amount?
(c) the conditional probability that the wife earns more than \) 25,000 given that the husband earns less than this amount?
Consider a sample of size drawn in the following manner: We start with an urn containing white and red balls. At each stage, a ball is drawn and its color is noted. The ball is then returned to the urn, along with an additional ball of the same color. Find the probability that the sample will contain exactly
(a) white balls;
(b) white ball;
(c) white balls;
(d) white balls.
In any given year, a male automobile policyholder will make a claim with probability pm and a female policyholder will make a claim with probability pf, where pf pm. The fraction of the policyholders that are male is α, 0 <α< 1. A policyholder is randomly chosen. If Ai denotes the event that this policyholder will make a claim in year i, show that P(A2|A1) > P(A1)
Give an intuitive explanation of why the preceding inequality is true.
If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events? Explain your reasoning.
(a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes.
(b) E is the event that a professor owns a car, and F is the event that he is listed in the telephone book.
(c) E is the event that a man is under 6 feet tall, and F is the event that he weighs more than 200 pounds.
(d) E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere.
(e) E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow.
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