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Chapter 3: Q. 3.19 (page 108)
Consider the gambler’s ruin problem, with the exception that and agree to play no more than n games. Let denote the probability that winds up with all the money when starts with and starts with . Derive an equation for in terms of and , and compute ,.
Value ofin terms ofandis
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Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our disposal is a biased coin that lands on heads with some unknown probability p that need not be equal to 1 2 . Consider the following procedure for accomplishing our task: 1. Flip the coin. 2. Flip the coin again. 3. If both flips land on heads or both land on tails, return to step 1. 4. Let the result of the last flip be the result of the experiment.
(a) Show that the result is equally likely to be either heads or tails.
(b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?
A true–false question is to be posed to a husband and-wife team on a quiz show. Both the husband and the wife will independently give the correct answer with probability p. Which of the following is a better strategy for the couple?
(a) Choose one of them and let that person answer the question.
(b) Have them both consider the question, and then either give the common answer if they agree or, if they disagree, flip a coin to determine which answer to give
In Example, what is the probability that someone has an accident in the second year given that he or she had no accidents in the first year?
A recent college graduate is planning to take the first three actuarial examinations in the coming summer. She will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is. If she passes the first exam, then the conditional probability that she passes the second one is , and if she passes both the first and the second exams, then the conditional probability that she passes the third exam is .7.
(a) What is the probability that she passes all three exams?
(b) Given that she did not pass all three exams, what is the conditional probability that she failed the second exam?
A town council of members contains a steering committee of size . New ideas for legislation go first to the steering committee and then on to the council as a whole if at least of the committee members approve the legislation. Once at the full council, the legislation requires a majority vote (of at least ) to pass. Consider a new piece of legislation, and suppose that each town council member will approve it, independently, with probability p. What is the probability that a given steering committee member’s vote is decisive in the sense that if that person’s vote were reversed, then the final fate of the legislation would be reversed? What is the corresponding probability for a given council member not on the steering committee?
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