Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 25 purchasers. a. What are the mean value and standard deviation of the number who want a new copy of the book? b. What is the probability that the number who want new copies is more than two standard deviations away from the mean value? c. The bookstore has 15 new copies and 15 used copies in stock. If 25 people come in one by one to purchase this text, what is the probability that all 25 will get the type of book they want from current stock? [Hint: Let X = the number who want a new copy. For what values of X will all 25 get what they want?] d. Suppose that new copies cost $100 and used copies cost$70. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 25 copies purchased? Be sure to indicate what rule of expected value you are using. [Hint: Let h(X) = the revenue when X of the 25 purchasers want new copies. Express this as a linear function.]

The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25 students who have recently taken the test. a. What is the probability that exactly 1 received a special accommodation? b. What is the probability that at least 1 received a special accommodation? c. What is the probability that at least 2 received a special accommodation? d. What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated? e. Suppose that a student who does not receive a special accommodation is allowed 3 hours for the exam, whereas an accommodated student is allowed 4.5 hours. What would you expect the average time allowed the 25 selected students to be?

Let X be the number of students who show up for a professor’s office hour on a particular day. Suppose that the pmf of X is $p(0) = .20, p(1) = .25, p(2) = .30, p(3) = .15,$ and $p(4) = .10$. What is the probability that at least two students show up? More than two students show up?

Let X be the number of students who show up for a professor’s office hour on a particular day. Suppose that the pmf of X is $p(0) = .20, p(1) = .25, p(2) = .30, p(3) = .15,$ and $p(4) = .10$. What is the probability that between one and three students, inclusive, show up?

A company that produces fine crystal knows from experience that 10% of its goblets have cosmetic flaws and must be classified as “seconds.” a. Among six randomly selected goblets, how likely is it that only one is a second? b. Among six randomly selected goblets, what is the probability that at least two are seconds? c. If goblets are examined one by one, what is the probability that at most five must be selected to find four that are not seconds?

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.

$$\begin{matrix} \text{y} & \text{45} & \text{46} & \text{47} & \text{48} & \text{49} & \text{50} & \text{51} & \text{52} & \text{53} & \text{54} & \text{55}\\ \text{p(y)} & \text{.05} & \text{.10} & \text{.12} & \text{.14} & \text{.25} & \text{.17} & \text{.06} & \text{.05} & \text{.03} & \text{.02} & \text{.01}\\ \end{matrix}$$

a. What is the probability that the flight will accommodate all ticketed passengers who show up? b. What is the probability that not all ticketed passengers who show up can be accommodated? c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis. a. What is the pmf of the number of granite specimens selected for analysis? b. What is the probability that all specimens of one of the two types of rock are selected for analysis? c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?

A particular type of tennis racket comes in a midsize version and an oversize version. Sixty percent of all customers at a certain store want the oversize version. Among ten randomly selected customers, what is the probability that the number who want the oversize version is within $1$ standard deviation of the mean value?

Suppose that p=P(male birth)=.5. A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled. a. What is the probability that the family has x male children? b. What is the probability that the family has four children? c. What is the probability that the family has at most four children? d. How many male children would you expect this family to have? How many children would you expect this family to have?

The number of requests for assistance received by a towing service is a Poisson process with rate $\alpha =4$ per hour. a. Compute the probability that exactly ten requests are received during a particular 2-hour period. b. If the operators of the towing service take a 30-min break for lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break?

The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous rv X with pdf

$$f(x)=\left\{\begin{array}{ll} k\left[1-(x-3)^{2}\right] & 2 \leq x \leq 4 \\ 0 & \mathrm { otherwise } \end{array}\right.$$

a. Sketch the graph of f(x). b. Find the value of k. c. What is the probability that the actual tracking weight is greater than the prescribed weight? d. What is the probability that the actual weight is within .25 g of the prescribed weight? e. What is the probability that the actual weight differs from the prescribed weight by more than .5 g?

Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices, with P(A)=.5, P(B)=.2, and P(C)=.3. a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card? Explain your reasoning. b. Answer part (a) for the number among the 100 who don’t pay with cash.

The number of people arriving for treatment at an emergency room can be modeled by a Poisson process with a rate parameter of five per hour. a. What is the probability that exactly four arrivals occur during a particular hour? b. What is the probability that at least four people arrive during a particular hour? c. How many people do you expect to arrive during a 45-min period?

A family decides to have children until it has three children of the same gender. Assuming P(B)=P(G)=.5, what is the pmf of X= the number of children in the family?