91Ó°ÊÓ

The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month. (a) Find the probability that, in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month. (b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides? (c) Counting the present month as month number \(1,\) what is the probability that the first month to have 8 or more suicides will be month number \(i, i \geq 1 ?\) What assumptions are you making?

An interviewer is given a list of people she can interview. If the interviewer needs to interview 5 people, and if each person (independently) agrees to be intervicwed with probability \(\frac{2}{3},\) what is the probability that her list of people will enable her to obtain her necessary number of intervicws if the list consists of (a) 5 people and (b) 8 people? For part (b), what is the probability that the interviewer will speak to exactly (c) 6 people and (d) 7 people on the list?

An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly \(n\) selections?

Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If \(X\) is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) \(P\\{X=0\\}\) and \((b) P\\{X>2\\}\)

A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 through \(80 .\) A player can select from 1 to 15 numbers; a win occurs if some fraction of the player's chosen subset matches any of the 20 numbers drawn by the house. The payoff is a function of the number of elements in the player's selection and the number of matches. For instance, if the player selects only 1 number, then he or she wins if this number is among the set of \(20,\) and the payoff is \(\$ 2.2\) won for every dollar bet. (As the player's probability of winning in this case is \(\frac{1}{4},\) it is clear that the "fair" payoff should be \(\$ 3\) won for every \(\$ 1\) bet.) When the player selects 2 numbers, a payoff (of odds) of \(\$ 12\) won for every \(\$ 1\) bet is made when both numbers are among the 20 (a) What would be the fair payoff in this case? Let \(P_{n, k}\) denote the probability that exactly \(k\) of the \(n\) numbers chosen by the player are among the 20 selected by the house. (b) Compute \(P_{n, k}\) (c) The most typical wager at Keno consists of selecting 10 numbers. For such a bet the casino pays off as shown in the following table. Compute the expected payoff: $$\begin{array}{cc} \hline \multicolumn{2}{c} {\text { Keno Payoffs in 10 Number Bets }} \\ \hline \text { Number of matches } & \text { Dollars won for each \$1 bet } \\\ \hline 0-4 & -1 \\ 5 & 1 \\ 6 & 17 \\ 7 & 179 \\ 8 & 1,299 \\ 9 & 2,599 \\ 10 & 24,999 \\ \hline \end{array}$$