Chapter 3: Q. 3.25 (page 109)
Prove directly that,
By applying the definition of conditional probability,the direction starting from the right side.
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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?
In any given year, a male automobile policyholder will make a claim with probability and a female policyholder will make a claim with probability localid="1646823185045" where . The fraction of the policyholders that are male is A policyholder is randomly chosen. If denotes the event that this policyholder will make a claim in the year show that
Give an intuitive explanation of why the preceding inequality is true.
With probability , the present was hidden by mom; with probability , it was hidden by dad. When mom hides the present, she hides it upstairs percent of the time and downstairs percent of the time. Dad is equally likely to hide it upstairs or downstairs.
(a) What is the probability that the present is upstairs?
(b) Given that it is downstairs, what is the probability it was hidden by dad?
A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves; it turns out to be an ace. The ace is then placed in the second half-deck. The half is then shuffled, and a card is drawn from it. Compute the probability that this drawn card is an ace. Hint: Condition on whether or not the interchanged card is selected
A total of percent of the voters in a certain city classify themselves as Independents, whereas percent classify themselves as Liberals and percent say that they are Conservatives. In a recent local election, percent of the Independents, percent of the Liberals, and percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election, what is the probability that he or she is
(a) an Independent?
(b) a Liberal?
(c) a Conservative?
(d) What percent of voters participated in the local election?
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