Math 210 Spring 2019

#### Problem Set 2, Due Tues. Feb. 5 in Class (late papers OK until 1:00 Wednesday)

Please put your papers in the mailbox (DRL 4W3) of our TA Yansong Gao

1. The next three players in a game win 30%, 20% and 25% of the time, respectively. What is the likelihood that none of them will win this time?
[Equivalent wording: It is the fifth inning of a baseball game. The batting averages of the next three batters are .300, .200, and .250. Say they face an average pitcher. What is the likelihood that none of them will get a hit this inning?]

2. A big dart board consists of three concentric disks of radius 1, 2, and 3 feet. If a dart lands in the center disk you get 50 points. If it lands in the middle ring you get 25 points, while if it lands in the outer ring you get 10 points.
Compute both the expected value and standard deviation of the number of points you'll get by throwing darts at random at this board.

3. a) In a certain town there are 2 bus companies whose buses stop at the Main Street station. One company's busses run every 8 minutes, the other runs every 10 minutes -- but the times of arrival of the previous busses are unknown. The question is, what is the average length of time that you will wait for a bus after arriving at the station?
b). Same as the above, but 3 bus companies -- whose busses each stop every 10 minutes.

4. You are interviewing 6 candidates for a job. As you proceed, you determine the relative ranks of the candidates (you won't know the "true rank" until you have interviewed all of them). Thus, if there are 6 candidates with true rank 6, 1, 4, 2, 3, 5, then after interviewing the first three candidates you would rank them 3, 1, 2. Alas, after you interview a candidate, you either hire that person or the candidate leaves and can no longer be considered.
You want a strategy when to stop and accept a candidate, maximizing the likelihood of getting the best candidate. Assume there are 6 candidates, and they arrive in a random order.
a) What is the probability that you get the best candidate if you interview all of the candidates? What if you immediately choose the first candidate?
b) Say you adopt the strategy of interviewing the first half of the candidates and then accept the first of the following candidates who is better than any seen so far (if you have seen all the candidates so are at the last candidate then by these rules you must accept that person). Show (by a crude estimate) that you have a chance of less than 50% of getting the the best candidate -- but better than a 25% chance of getting the the best candidate.
SOLUTION: Using this strategy you win if the second best candidate is in the first group of 5 and the best candidate is in the last group of 5, so 25% of the time.

5. In the Cancer Test Case, we used that everyone who has the cancer tests positive. Instead, say 5% of those who have the cancer test negative (false negative). Of those who test negative, what is the likelihood that they have this cancer?

6. [Monty Hall Problem]   Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch?
What should you do -- and why? A "tree diagram" may be useful.

7. In an oft-cited experiment the psychologists Kahneman and Tversky asked a group of subjects to imagine the outbreak of an unusual disease, expected to kill 600 people, and to choose between two public health programs to combat it.
Program A, the subjects were told, had 400 people would die.
Program B had a one-third probability that no one would die and a two-thirds probability that 600 people would die
Which would you choose? Explain your decision.

8. a). Would you accept a gamble that offers a 10-percent chance to win \$95 and a 90-percent chance to lose \$5?
b). Would you pay \$5 to participate in a lottery that offers a 10-percent chance to win \$100 and a 90-percent chance to win nothing?
c). Do the options in a)-b) offer identical outcomes?
Remarks:
a). The great majority of people in the study rejected this proposition as a loser.
b)-c). A large proportion of those who refused the first option accepted the second. But the options offer identical outcomes. As Kahneman and Tversky see it: "Thinking of the \$5 as a payment makes the venture more acceptable than thinking of the same amount as a loss." It's all a matter of how the situation is framed in this case, the extent to which people are "risk averse."

B-1. [Blood Test] A number, k, of people are subjected to a blood test, the result of which is either "positive" or "negative". It can be processed in two ways:
i). Each person can be tested separately, so k tests are required.
ii). The blood samples of all k people can be pooled and analyzed together. If this test is negative, then one test suffices for the k people, while if the test is positive, each of the k people must be tested separately so k+1 tests are then required.
Assume that the probability, p, that a test is positive is the same for all people and that these events are all independent.
a). Find the probability that the test for a pooled sample of k people will be positive.
b). What is the expected value of the number of tests necessary under plan ii).?

SOLUTION: ExpectedVal =1(1-p)k + (k+1)[1-(1-p)k]
c). A large number, N, of people are subjected to a blood test, the result of which is either "positive" or "negative", just as above. The plan is to run the blood tests in batches of k people [assume that N is divisible by k].
Compute the expected value of the number of blood tests needed.
SOLUTION: ExpectedVal = (N/k){1(1-p)k + (k+1)[1-(1-p)k]}  =  N[1 - (1-p)k + (1/k)]

d). For small p, show that the value of k that will minimize the expected number of tests made under plan ii). is approximately 1/√p.
SOLUTION Take the derivatve with respect to k.
Reference: Pooled Tests