Nathan Cheboygan, a singing gambler from northern Michigan, is famous for his loaded dice. The following table shows the probability distribution for the sum, denoted by $x$, of the faces on a pair of Nathan's dice.

x	2	3	4	5	6	7	8	9	10	11	12
P(x)	.065	.065	.08	.095	.11	.17	.11	.095	.08	.065	.065

a. Draw a bar graph for this probability distribution.

b. Determine the probability that the sum of the faces on a single roll of Nathan's dice is

i. an even number

ii. 7 or 11

iii. 4 to 6

iv. no less than 9

Let P(x), Q(x), and R(x) be the statements “x is a professor,” “x is ignorant,” and “x is vain,” respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people. a) No professors are ignorant. b) All ignorant people are vain. c) No professors are vain. d) Does (c) follow from (a) and (b)?

Let P (m, n) be the statement “m divides n,” where the domain for both variables consists of all positive integers. (By “m divides n” we mean that n = km for some integer k.) Determine the truth values of each of these statements. a) P (4, 5) b) P (2, 4) c) ∀m ∀n P (m, n) d) ∃m ∀n P (m, n) e) ∃n ∀m P (m, n) f ) ∀n P (1, n)

The Poisson probability distribution's parameter or parameters is/are a. $\lambda$ b. $\lambda$ and $x$ c. $\lambda$ and $e$

Let R₁ and R₂ be the “divides” and “is a multiple of” relations on the set of all positive integers, respectively. That is, R₁ = {(a, b) | a divides b} and R₂ = {(a, b) | a is a multiple of b}. Find a) R₁ ∪ R₂. b) R₁ ∩ R₂. c) R₁ − R₂. d) R₂ − R₁. e) R₁ ⊕ R₂.

Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 3), (2, 4), (3, 1), (3, 5), (4, 3), (5, 1), (5, 2), and (5, 4). Find R^2 , R^3 , R^4 , R^5 , and R^6 .

In this chapter's Uses and Misuses section, we learned how to determine the odds of various poker hands. Find the probability of being dealt a. three of a kind b. two pairs c. one pair

A supermarket's checkout counter sees 10 customers each hour on average, and these arrivals follow a Poisson distribution. Find the likelihood that exactly 4 consumers will visit this checkout counter over the course of a 2-hour period using each of the next two techniques. a. Use the arrivals in each of the two nonoverlapping 1-hour periods and then add these. (Note that the numbers of arrivals in two nonoverlapping periods are independent of each other.) b. Use the arrivals in a single 2-hour period.

You've undoubtedly all played the "Rock, Paper, Scissors" game as kids. Take this game's variation into consideration. Let's say there are three players instead of just two, and we'll refer to them as A, B, and C. Independent of one another, each player chooses one of the three options, Rock, Paper, or Scissors. If all three players choose the same thing—say, a rock—the game belongs to Player A. If exactly two of the three players choose the same object and the third player chooses a different item, Player B will win the game. Player $\mathrm{C}$ will win the game if every player selects a different item. If Player B wins the game, he or she will be paid $1. If Player$ $\mathrm{C} wins the game, he or she will be paid$3. How much should Player A be paid if they win the game, assuming that the projected earnings should be the same for each player to make this a fair game?

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 20 per hour. The manager notices that within a randomly chosen 15-minute period, 9 calls were received by the mail-order business. a. What is the likelihood that nine or more calls will come in within a particular 15-minute period if the real call rate is 20 per hour? b. What is the likelihood that 9 or more calls will come in within a particular 15-minute period if the call rate is actually 30 per hour? c. Do you believe that the rate of incoming calls is more likely to be 20 or 30 per hour, based on the calculations in parts a and b? d. Should the manager bring on a second operator, in your opinion? Explain.

Brad Henry is a salesperson for stone products. Let $x$ represent the total number of contacts he meets with on a given day. The probability distribution for $x$ is provided in the table below.

$$\begin{array}{l|cccc} \hline x & 1 & 2 & 3 & 4 \\ \hline P(x) & .12 & .25 & .56 & .07 \\ \hline \end{array}$$

Let $y$ represent the total number of people Brad meets on two days chosen at random. For $y$, jot down the probability distribution.

Think about the next three games. Which one do you think you'll play the most? Which one do you think you'd play the least? Describe your response mathematically.

Game I: You toss a fair coin once. If a head appears you receive $3, but if a tail appears you have to pay$1. Game II: You buy a single ticket for a raffle that has a total of 500 tickets. Two tickets are chosen without replacement from the 500 . The holder of the first ticket selected receives $300, and the holder of the second ticket selected receives$150. Game III: You toss a fair coin once. If a head appears you receive $1,000,002, but if a tail appears you have to pay$1,000,000.

A history teacher has given her class a list of seven essay questions to study before the next test. The teacher announced that she will choose four of the seven questions to give on the test, and each student will have to answer three of those four questions. a. How many different options are there for the teacher to select four of the seven questions? b. Assume a student only has time to study five questions. How many times can the teacher choose four of the seven possible questions, making sure that two of them are ones that the student did not study? c. What is the likelihood that a student in part b will be required to respond to a question that they did not prepare for? What are the chances that the two questions the student didn't study would be among the four on the test?

Suppose that a certain casino has the "money wheel" game. The money wheel is divided into 50 sections, and the wheel has an equal probability of stopping on each of the 50 sections when it is spun. On this wheel, 22 of the sections display $1 bills, 14 display$2 bills, 7 display $5 bills, 3 display$10 bills, 2 display $20 bills, I display a flag, and 1 display a joker. One can wager on any of the seven potential outcomes. The bettor wins if the outcome they chose came to pass when the wheel stopped. The following are the net payouts for these results for wagers of$1.

$$\begin{array}{l|rrrrrrr} \hline \text { Symbol bet on } & \$ 1 & \$ 2 & \$ 5 & \$ 10 & \$ 20 & \text { Flag } & \text { Joker } \\ \hline \text { Payoff (dollars) } & 1 & 2 & 5 & 10 & 20 & 40 & 40 \\ \hline \end{array}$$

a. What is the estimated net return if the gambler stakes $1 on the result? b. Determine the predicted net payoffs for every one of the remaining six scenarios. c. Which wagers are the best in terms of predicted net return? What is/are the worst?