In a primary election, there are four candidates for mayor, five candidates for city treasurer, and two candidates for county attorney. (a) In how many ways can a voter mark his ballot for all three of these offices? (b) In how many ways can a person vote if he exercises his option of not voting for a candidate for any or all of these offices?

If the NCAA has applications from six universities for hosting its intercollegiate tennis championships in two consecutive years, in how many ways can they select the hosts for these championships (a) if they are not both to be held at the same university; (b) if they may both be held at the same university?

Counting the number of outcomes in games of chance has been a popular pastime for many centuries. This was of interest not only because of the gambling that was involved but also because the outcomes of games of chance were often interpreted as divine intent. Thus, it was just about a thousand years ago that a bishop in what is now Belgium determined that there are 56 different ways in which three dice can fall provided one is interested only in the overall result and not in which die does what. He assigned a virtue to each of these possibilities and each sinner had to concentrate for some time on the virtue that corresponded to his cast of the dice. (a) Find the number of ways in which three dice can all come up with the same number of points. (b) Find the number of ways in which two of the three dice can come up with the same number of points, while the third comes up with a different number of points. (c) Find the number of ways in which all three of the dice can come up with a different number of points. (d) Use the results of parts (a), (b), and (c) to verify the bishop's calculations that there are altogether 56 possibilities.

Extending the Addition Rule Extend the formal addition rule to develop an expression for $P(A$ or $B$ or $C)$.

Suppose that A and B are mutually exclusive events. a. Use the special addition rule to express P(A or B) in terms of P(A) and P(B). b. Show that the general addition rule gives the same answer as that in part (a).

The probability of surviving a certain transplant operation is 0.55. If a patient survives the operation, the probability that his or her body will reject the transplant within a month is 0.20. What is the probability of surviving both of these critical stages?

Find the number of possible 5-card hands that contain the cards specified.

5 face cards (either kings, queens, or jacks)

A person with $\$ 2$ in her pocket bets $\$ 1$, even money, on the flip of a coin, and she continues to bet $\$ 1$ as long as she has any money. Draw a tree diagram to show the various things that can happen during the first four flips of the coin. After the fourth flip of the coin, in how many of the cases will she be (a) exactly even; (b) exactly $\$ 2$ ahead?

From a penny, a nickel, a dime, and a quarter, how many different sums of money can be formed consisting of a) three coins? b) at most two coins?

Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let X=the number among the next 15 purchasers who select the chain-driven model. a. What is the pmf of X? b. Compute P(X>10). c. Compute $P(6 \leq X \leq 10)$. d. Compute $\mu$ and $\sigma ^{2}$. e. If the store currently has in stock 10 chain-driven models and 8 shaft-driven models, what is the probability that the requests of these 15 customers can all be met from existing stock?

A particular iPod playlist contains 100 songs, of which 10 are by the Beatles. Suppose the shuffle feature is used to play the songs in random order (the randomness of the shuffling process is investigated in "Does Your iPod Really Play Favorites?" (The Amer. Statistician, 2009: 263 - 268)). What is the probability that the first Beatles song heard is the fifth song played?

An urn contains twenty chips, numbered 1 through 20. Two are drawn simultaneously. What is the probability that the numbers on the two chips will differ by more than 2?

A soft-drink machine at a steak house is regulated so that the amount of drink dispensed is approximately normally distributed with a mean of 200 milliliters and a standard deviation of 15 milliliters. The machine is checked periodically by taking a sample of 9 drinks and computing the average content. If $\overline{x}$ falls in the interval $191<\overline{x}<209,$ the machine is thought to be operating satisfactorily; otherwise, we conclude that $\mu \neq 200$ milliliters. (a) Find the probability of committing a type I error when $\mu=200$ milliliters. b. Find the probability of committing a type II error when $\mu=215$ milliliters.

A soft drink machine can be regulated to discharge an average of $\mu$ ounces per cup. If the ounces of fill are normally distributed, with a standard deviation equal to .3 ounce, give the setting for $\mu$ so that 8-ounce cups will overflow only 1% of the time.