Chapter 1: Problem 7
Let \(X\) have the pdf \(f(x)=3 x^{2}, 0
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List all possible arrangements of the four letters \(m, a, r\), and \(y .\) Let \(C_{1}\) be the collection of the arrangements in which \(y\) is in the last position. Let \(C_{2}\) be the collection of the arrangements in which \(m\) is in the first position. Find the union and the intersection of \(C_{1}\) and \(C_{2}\).
A die is cast independently until the first 6 appears. If the casting stops on an odd number of times, Bob wins; otherwise, Joe wins. (a) Assuming the die is fair, what is the probability that Bob wins? (b) Let \(p\) denote the probability of a 6 . Show that the game favors Bob, for all \(p\), \(0
A pair of dice is cast until either the sum of seven or eight appears. (a) Show that the probability of a seven before an eight is \(6 / 11\). (b) Next, this pair of dice is cast until a seven appears twice or until each of a six and eight has appeared at least once. Show that the probability of the six and eight occurring before two sevens is \(0.546\).
A coin is to be tossed as many times as necessary to turn up one head. Thus the elements \(c\) of the sample space \(\mathcal{C}\) are \(H, T H, T T H\), TTTH, and so forth. Let the probability set function \(P\) assign to these elements the respective probabilities \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\), and so forth. Show that \(P(\mathcal{C})=1 .\) Let \(C_{1}=\\{c:\) c is \(H, T H, T T H\), TTTH, or TTTTH \(\\}\). Compute \(P\left(C_{1}\right)\). Next, suppose that \(C_{2}=\) \(\\{c: c\) is TTTTH or TTTTTH \(\\}\). Compute \(P\left(C_{2}\right), P\left(C_{1} \cap C_{2}\right)\), and \(P\left(C_{1} \cup C_{2}\right)\).
Let \(X\) equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of \(X\) and compute the probability that \(X\) is equal to an odd number.
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