91Ó°ÊÓ

Towns A, B, C, and D are located in such a way that there are four roads from A to B, five roads from B to C, and six roads from C to D. How many routes are there from town A to town D via towns B and C?

Roulette In the game of roulette as played in Las Vegas, the wheel has 38 slots: Two slots are numbered 0 and 00 , and the rest are numbered 1 to \(36 .\) A \(\$ 1\) bet on any number other than 0 or 00 wins \(\$ 36\) (\$35 plus the \(\$ 1\) bet). Find the expected value of this game.

In a family of four children, how many different boy-girl birth-order combinations are possible? (The birth orders BBBG and BBGB are different.)

Sweepstakes \(A\) sweepstakes offers a first prize of \(\$ 1,000,000,\) second prize of \(\$ 100,000,\) and third prize of \(\$ 10,000\) . Suppose that two million people enter the contest and three names are drawn randomly for the three prizes. (a) Find the expected winnings for a person participating in this contest. (b) Is it worth paying a dollar to enter this sweepstakes?

7–12 Find the number of distinguishable permutations of the given letters. $$X X Y Y Z Z Z$$

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At most three failures

A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. Five hearts

Rolling Dice Six dice are rolled. Find the probability that 2 of them show a four.

13–18 Evaluate the expression. $$C(8,3)$$

A Game of Chance \(A\) box contains 100 envelopes. Ten envelopes contain \(\$ 10\) each, ten contain \(\$ 5\) each, two are "unlucky," and the rest are empty. A player draws an envelope from the box and keeps whatever is in it. If a person draws an unlucky envelope, however, he must pay \(\$ 100\) . What is the expectation of a person playing this game?