91Ó°ÊÓ

For a finite set \(A\), let \(N(A)\) denote the number of elements in \(A\). (a) Show that $$ N(A \cup B)=N(A)+N(B)-N(A B) $$ (b) More generally, show that $$ N\left(\bigcup_{i=1}^{n} A_{i}\right)=\sum_{i} N\left(A_{i}\right)-\sum_{R

Suppose that an experiment is performed \(n\) times. For any event \(E\) of the sample space, let \(n(E)\) denote the number of times that event \(E\) occurs, and define \(f(E)=n(E) / n\). Show that \(f(\cdot)\) satisfies Axioms 1,2 , and 3 .

Consider an experiment that consists of six horses, numbered 1 through 6 , running a race and suppose that the sample space consists of the \(6 !\) possible orders in which the horses finish. Let \(A\) be the event that the number 1 horse is among the top three finishers, and let \(B\) be the event that the number 2 horse comes in second. How many outcomes are in the event \(A \cup B\) ?

Show that the probability that exactly one of the events \(E\) or \(F\) occurs equals \(P(E)+P(F)-2 P(E F)\)

Prove that \(P\left(E F^{c}\right)=P(E)-P(E F)\)

Prove Boole's inequality: $$ P\left(\bigcup_{i=1}^{\infty} A_{i}\right) \leq \sum_{i=1}^{\infty} P\left(A_{i}\right) $$

An um contains \(M\) white and \(N\) black balls. If a random sample of size \(r\) is chosen, what is the probability that it contains exactly \(k\) white balls?

Poker dice is played by simultaneously rolling 5 dice. Show that (a) \(P\\{\) no two alike \(\\}=.0926\) (b) \(P\) (one pair \(\\}=.4630\) (c) \(P\\{\) two pair \(\\}=.2315\) (d) \(P\\{\) three alike \(\\}=.1543\); (e) \(P\\{\) full house \(\\}=.0386\); (f) \(P\) (four alike \(\\}=0193\) (g) \(P\\{\) five alike \(\\}=.0008\).

Consider the matching problem, Example \(5 \mathrm{~m}\), and define \(A_{N}\) to be the number of ways in which the \(N\) men can select their hats so that no man selects his own. Argue that $$ A_{N}=(N-1)\left(A_{N-1}+A_{N-2}\right) $$ This formula, along with the boundary conditions \(A_{1}=0, A_{2}=1\), can then be solved for \(A_{N}\), and the desired probability of no matches would be \(A_{N} / N !\) HINT: After the first man selects a hat that is not his own, there remain \(N-1\) men to select among a set of \(N-1\) hats that does not contain the hat of one of these men. Thus there is one extra man and one extra hat. Argue that we can get no matches cither with the extra man selecting the extra hat or with the extra man not selecting the extra hat.

Let \(f_{n}\) denote the number of ways of tossing a coin \(n\) times such that successive heads never appear. Argue that $$ f_{n}=f_{n-1}+f_{n-2} \quad n \geq 2, \text { where } f_{0} \equiv 1, f_{1} \equiv 2 $$ HINT: How many outcomes are there that start with a head, and how many start with a tail? If \(P_{n}\) denotes the probability that successive heads never appear when a coin is tossed \(n\) times, find \(P_{n}\) (in terms of \(f_{n}\) ) when all possible outcomes of the \(n\) tosses are assumed equally likely. Compute \(P_{10}\).