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Chapter 4: Problem 5

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 \(X ?\)

Short Answer

Expert verified
For even values of \(n\), the possible values of \(X\) are: \[X \in \{ 0, 2, 4, \dots , n \}\] For odd values of \(n\), the possible values of \(X\) are: \[X \in \{ 1, 3, 5, \dots , n \}\]

Step by step solution

01

Identifying the minimum difference

: In the minimum difference case, the number of heads and the number of tails would be equal. This can happen only when we have an even number of tosses (n is even). In this case, we can have half heads and half tails, which makes the minimum difference 0 (zero). If n is odd, it is impossible to have an equal number of heads and tails, so the minimum difference would be 1.
02

Identifying the maximum difference

: The maximum difference between the number of heads and the number of tails would occur when either all tosses result in heads or all tosses result in tails. In this case, X would be equal to the total number of tosses, n.
03

Listing possible values of X

: Now that we know the bounds of X, we can list all possible values within these bounds. For even values of n, the minimum difference in heads and tails is 0, and the maximum difference is n. For odd values of n, the minimum difference is 1, and the maximum difference is still n. In each case, the difference will always be an even number. So, for an even value of n, the possible values of X are: \[x \in \{ 0, 2, 4, \dots , n \}\] For an odd value of n, the possible values of X are: \[x \in \{ 1, 3, 5, \dots , n \}\]

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Most popular questions from this chapter

From a set of \(n\) randomly chosen people let \(E_{i j}\) denote the event that persons \(i\) and \(j\) have the same birthday. Assume that each person is equally likely to have any of the 365 days of the year as his or her birthday. Find (a) \(P\left(E_{3,4} \mid E_{1,2}\right)\); (b) \(P\left(E_{1,3} \mid E_{1,2}\right) ;\) (c) \(P\left(E_{2,3} \mid E_{1,2} \cap E_{1,3}\right)\). What can you conclude from the above about the independence of the \(\left(\begin{array}{l}n \\ 2\end{array}\right)\) events \(E_{i j} ?\)

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 with 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:

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Compare the Poisson approximation with the correct binomial probability for the following cases: (a) \(P\\{X=2\\}\) when \(n=8, p=.1\); (b) \(P\\{X=9\\}\) when \(n=10, p=.95\); (c) \(P\\{X=0\\}\) when \(n=10, p=.1\); (d) \(P\\{X=4\\}\) when \(n=9, p=.2\).

For a nonnegative integer-valued random variable \(N\), show that $$ E[N]=\sum_{i=1}^{\infty} P\\{N \geq i\\} $$
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