91Ó°ÊÓ

Gerolamo Cardano in his book, The Gambling Scholar, written in the early 1500 s, considers the following carnival game. There are six dice. Each of the dice has five blank sides. The sixth side has a number between 1 and 6 -a different number on each die. The six dice are rolled and the player wins a prize depending on the total of the numbers which turn up. (a) Find, as Cardano did, the expected total without finding its distribution. (b) Large prizes were given for large totals with a modest fee to play the game. Explain why this could be done.

Exactly one of six similar keys opens a certain door. If you try the keys, one after another, what is the expected number of keys that you will have to try before success?

Let \(X\) be a random variable with \(E(X)=\mu\) and \(V(X)=\sigma^{2}\). Show that the function \(f(x)\) defined by $$f(x)=\sum_{\omega}(X(\omega)-x)^{2} p(\omega)$$ has its minimum value when \(x=\mu\).

A professor wishes to make up a true-false exam with \(n\) questions. She assumes that she can design the problems in such a way that a student will answer the \(j\) th problem correctly with probability \(p_{j},\) and that the answers to the various problems may be considered independent experiments. Let \(S_{n}\) be the number of problems that a student will get correct. The professor wishes to choose \(p_{j}\) so that \(E\left(S_{n}\right)=.7 n\) and so that the variance of \(S_{n}\) is as large as possible. Show that, to achieve this, she should choose \(p_{j}=.7\) for all \(j\); that is, she should make all the problems have the same difficulty.