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Chapter 2: Problem 2
Let \(X\) represent the difference between the number of heads and the number of tails obtained when a coin is tossed \(n\) times. What are the possible values of \(X\) ?
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Suppose that two teams are playing a series of games, each of which is independently won by team \(A\) with probability \(p\) and by team \(B\) with probability \(1-p\). The winner of the series is the first team to win 4 games. Find the expected number of games that are played, and evaluate this quantity when \(p=1 / 2\).
In Exercise 2, if the coin is assumed fair, then, for \(n=2\), what are the probabilities associated with the values that \(X\) can take on?
Let \(X\) denote the number of white balls selected when \(k\) balls are chosen at random from an urn containing \(n\) white and \(m\) black balls.
Suppose \(X\) has a binomial distribution with parameters 6 and \(\frac{1}{2}\). Show that \(X=3\) is the most likely outcome.
Suppose that an expcriment can result in one of \(r\) possible outcomes, the \(i\) th outcome having probability \(p_{i+} i=1, \ldots, r, \sum_{i=1}^{\prime} p_{i}=1 .\) If \(n\) of these experiments are performed, and if the outcome of any one of the \(n\) does not affect the outcome of the other \(n-1\) experiments, then show that the probability that the first outcome appears \(x_{1}\) times, the second \(x_{2}\) times, and the \(r\) th \(x\), times is $$ \frac{n !}{x_{1}\left\lfloor x_{2} ! \cdots x_{r} !\right.} p_{1}^{x} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \quad \text { when } x_{1}+x_{2}+\cdots+x_{r}=n $$ This is known as the multinomial distribution.
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