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A coin having probability \(p\) of coming up heads is successively flipped until two of the most recent three flips are heads. Let \(N\) denote the number of flips. (Note that if the first two flips are heads, then \(N=2\).) Find \(E[N]\).

A coin, having probability \(p\) of landing heads, is continually flipped until at least one head and one tail have been flipped. (a) Find the expected number of flips needed. (b) Find the expected number of flips that land on heads. (c) Find the expected number of flips that land on tails. (d) Repeat part (a) in the case where flipping is continued until a total of at least two heads and one tail have been flipped.

A gambler wins each game with probability \(p .\) In each of the following cases, determine the expected total number of wins. (a) The gambler will play \(n\) games; if he wins \(X\) of these games, then he will play an additional \(X\) games before stopping. (b) The gambler will play until he wins; if it takes him \(Y\) games to get this win, then he will play an additional \(Y\) games.

A coin that comes up heads with probability \(p\) is continually flipped until the pattern \(\mathrm{T}, \mathrm{T}\), \(\mathrm{H}\) appears. (That is, you stop flipping when the most recent flip lands heads, , ind the two immediately preceding it lands tails.) Let \(X\) denote the number of flips made, and find \(E[X]\).

A set of \(n\) dice is thrown. All those that land on six are put aside, and the others are again thrown. This is repeated until all the dice have landed on six. Let \(N\) denote the number of throws needed. (For instance, suppose that \(n=3\) and that on the initial throw exactly two of the dice land on six. Then the other die will be thrown, and if it lands on six, then \(N=2\).) Let \(m_{n}=E[N]\). (a) Derive a recursive formula for \(m_{n}\) and use it to calculate \(m_{i}, i=2,3,4\) and to show that \(m_{5} \approx 13.024\). (b) Let \(X_{i}\) denote the number of dice rolled on the \(i\) th throw. Find \(E\left[\sum_{i=1}^{N} X_{i}\right]\).

A manuscript is sent to a typing firm consisting of typists \(A, B\), and \(C\). If it is typed by \(A\), then the number of errors made is a Poisson random variable with meap \(2.6\); if typed by \(B\), then the number of errors is a Poisson random variable with mean 3 ; and if typed by \(C\), then it is a Poisson random variable with mean 3.4. Let \(X\) denote the number of errors in the typed manuscript. Assume that each typist is equally likely to do the work. (a) Find \(E[X]\). (b) Find \(\operatorname{Var}(X)\).

A prisoner is trapped in a cell containing three doors. The first door leads to a tunnel that returns him to his cell after two days of travel. The second leads to a tunnel that returns him to his cell after three days of travel. The third door leads immediately to freedom. (a) Assuming that the prisoner will always select doors 1,2, and 3 with probabilities \(0.5,0.3,0.2\), what is the expected number of days until he reaches freedom? (b) Assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (In this version, for instance, if the prisoner initially tries door 1 , then when he returns to the cell, he will now select only from doors 2 and 3.) (c) For parts (a) and (b) find the variance of the number of days until the prisoner reaches freedom.

A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to the right, then it will wander around in the maze for three minutes and will then return to its initial position. If it goes to the left, then with probability \(\frac{1}{3}\) it will depart the maze after two minutes of traveling, and with probability \(\frac{2}{3}\) it will return to its initial position after five minutes of traveling. Assuming that the rat is at all times equally likely to go to the left or the right, what is the expected number of minutes that it will be trapped in the maze?

The number of customers entering a store on a given day is Poisson distributed with mean \(\lambda=10\). The amount of money spent by a customer is uniformly distributed over \((0,100)\). Find the mean and variance of the amount of money that the store takes in on a given day.

A and B play a series of games with A winning each game with probability \(p\). The overall winner is the first player to have won two more games than the other. (a) Find the probability that \(\mathrm{A}\) is the overall winner. (b) Find the expected number of games played.