91Ó°ÊÓ

In Problem \(5,\) for \(n=3,\) if the coin is assumed fair, what are the probabilities associated with the values that \(X\) can take on?

In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultancously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specificd player, and denote by \(X\) the amount of money he wins in a single game of Two-Finger Morra. (a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of \(X\) and what are their associated probabilities? (b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1 or 2 fingers, what are the possible values of \(X\) and their associated probabilities?

Four independent flips of a fair coin are made. Let \(X\) denote the number of heads obtained. Plot the probability mass function of the random variable \(x-2\)

One of the numbers 1 through 10 is randomly chosen. You are to try to guess the number chosen by asking questions with "yes-no" answers. Compute the expected number of questions you will need to ask in each of the following two cases: (a) Your ith question is to be "Is it i?" \(i=\) 1,2,3,4,5,6,7,8,9,10 (b) With each question you try to eliminate onehalf of the remaining numbers, as nearly as possible.

A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct. What is the probability that he would have done at least this well if he had no ESP?

Suppose that, in flight, airplane engines will fail with probability \(1-p,\) independently from engine to engine. If an airplane needs a majority of its engines operative to complete a successful flight, for what values of \(p\) is a 5 -engine plane preferable to a 3 -engine plane?

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter \(\lambda=3\) (a) Find the probability that 3 or more accidents occur today. (b) Repeat part (a) under the assumption that at least 1 accident occurs today.

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is \(\frac{1}{100},\) what is the (approximate) probability that you will win a prize (a) at least once? (b) exactly once? (c) at least twice?

Consider \(n\) independent trials, each of which results in one of the outcomes \(1, \ldots, k\) with respective probabilities \(p_{1}, \ldots, p_{k}, \quad \sum_{i=1}^{k} p_{i}=1 .\) Show that if all the \(p_{i}\) are small, then the probability that no trial outcome occurs more than once is approximately equal to \(\exp \left(-n(n-1) \sum_{i} p_{i}^{2} / 2\right)\)

People enter a gambling casino at a rate of 1 every 2 minutes. (a) What is the probability that no one enters between 12: 00 and \(12: 05 ?\) (b) What is the probability that at least 4 people enter the casino during that time?