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Chapter 3: Q. 3.1 (page 106)
Show that if , then
We proved that by applying conditional probability as.
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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?
Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and bad risks. The company鈥檚 records indicate that the probabilities that good-, average-, and bad-risk persons will be involved in an accident over a 1-year span are, respectively, .05, .15, and .30. If 20 percent of the population is a good risk, 50 percent an average risk, and 30 percent a bad risk, what proportion of people have accidents in a fixed year? If policyholder A had no accidents in 2012, what is the probability that he or she is a good risk? is an average risk?
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 伪, 0 <伪< 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.
A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate, and weak are .7, .2, and .1, respectively.
(a) How certain is she that she will receive the new job offer?
(b) Given that she does receive the offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?
(c) Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?
A total of married working couples were polled about their annual salaries , with the following information resulting:
| Wife | Husband | |
| Less than | More than | |
| Less than | ||
| More than | ||
For instance, in of the couples, the wife earned more and the husband earned less than \(. If one of the couples is randomly chosen, what is
(a) the probability that the husband earns less than \)?
(b) the conditional probability that the wife earns more than \(given that the husband earns more than this amount?
(c) the conditional probability that the wife earns more than \)given that the husband earns less than this amount?
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