Question

The college suspects their star player James Strongarm has used an illegal performance-enhancing drug. Currently, the probability that James is a drug user is 0.37. They consider barring him from athletics. The real "monetary" values of the outcomes are difficult to assess in this case and, therefore, the value of each outcome is expressed in a funny money unit: Galleon (from Harry Potter Movies). The following are the payoffs for the potential outcomes:

Correctly identifying a drug user and barring this person from athletics: 40 Galleon

Falsely accusing a nonuser and barring this person from athletics: -60 Galleon

Not identifying a drug user and allowing this person to participate in athletics: -30 Galleon

Not barring a nonuser: 0 Galleon

Before they make the decision, they can ask for a drug test, which costs -1 Galleon per single athlete . In the drug testing, assume there are two possible test results: positive and negative. For a drug user, the probabilities of these outcomes are 0.95 and 0.05, respectively. For a nonuser, they are 0.01 and 0.99, respectively. Assume that 2% of all college athletes are drug users.

(When you use the probabilities from the flipped table, use cell references in the decision tree instead of typing the probabilities manually. All choices in the following questions are based on the decision tree built using this practice.)

Question 9 (1 point)

If the test costs -20, what is the best decision alternative?

A Forgo the test and bar him.

B Forgo the test and do not bar him.

C Take the test. Bar him if the result is positive; do not bar him if the result is negative.

D Take the test, and do not bar him.

Question 10 (1 point)

The test costs -1. If the probability estimated by the college that James is a drug user is 0.60, what is the best decision alternative?

A Forgo the test and bar him.

B Forgo the test and do not bar him.

C Take the test. Bar him if the result is positive; do not bar him if the result is negative.

D Take the test, and do not bar him.

Question 11 (1 point)

What should be the maximum price of the drug test? (Assume that the probability that James is a drug user is 0.37.)

A 10.25

B 11.25

C 0.15

D 15.25

Answer 1

Baye's rule:

Positive
Prior		Conditional		Joint		Posterior
P(User)	0.02	P(Positive|User)	0.95	P(U) x P(Pos|U)	0.02	P(User|Positive)	0.66
P(Nonuser)	0.98	P(Positive|Nonuser)	0.01	P(N) x P(Pos|N)	0.01	P(Nonuser|Positive)	0.34
Total	1			P(Positive)	0.03		1.00
Negative
Prior		Conditional		Joint		Posterior
P(User)	0.02	P(Negative|User)	0.05	P(U) x P(Neg|U)	0.0010	P(User|Negative)	0.0010
P(Nonuser)	0.98	P(Negative|Nonuser)	0.99	P(N) x P(Neg|N)	0.9702	P(Nonuser|Negative)	0.9990
Total	1			P(Negative)	0.97		1.00

Q.9

Answer: B Forgo the test and do not bar him.

Decision tree

0.66 User 20 Identify & bar 40 20 0 -14 0.34 Nonuser 0.03 Positive -80 -60 -80 1 -14 0 0.66 User 50 Don't identify -30 -50 0 -39.8 0.34 Nonuser -20 Test 0 -20 -20 -19.8491 0.001 User Identify & bar 40 20 -79.9 0.999 Nonuser |0.97 Negative -80 60 -80 -20.03 0 0.001 User -50 Don't identify -30 -50 2 -11.1 0.999 0 -20.03 Nonuser -20 0 -20 0.37 User Identify & bar 40 -23 0.63 Nonuser -60 No test -60 -60 0 -11.1 0.37 User -30 Don't identify -30 -30 -11.1 0.63 Nonuser 0 0 40 20 40

Q.10

Answer: A Forgo the test and bar him.

Decision tree

0.66 User 39 Identify & bar 40 39 0 0.34 Nonuser 5 -61 0.03 Positive -60 -61 1 0 0.66 User 5 -31 Don't identify -30 -31 0 -20.8 0.34 Nonuser Test 0 -1 -1 -0.8491 0.001 User Identify & bar 40 39 -60.9 0.999 Nonuser |0.97 Negative -61 60 -61 0 -1.03 0.001 User -31 Don't identify -30 -31 2 0.999 0 -1.03 Nonuser -1 0 -1 0.6 User Identify & bar 40 0 0 0 4 Nonuser -60 No test -60 -60 1 0 0.6 User -30 Don't identify -30 -30 -18 0.4 Nonuser 0 0 40 39 40

Q.11

Answer: 0.1509 - (-11.1) = 11.25

0.66 User 40 Identify & bar 40 40 0 6 0.34 Nonuser -60 0.03 Positive -60 -60 1 6 0 0.66 User -30 Don't identify -30 -30 0 -19.8 0.34 Nonuser 0 Test 0 0 0.1509 0.001 User Identify & bar 40 -59.9 0.999 Nonuser |0.97 Negative -60 -60 60 2 0 -0.03 0.001 User -30 Don't identify -30 -30 0.1509 0.999 0 -0.03 Nonuser 0 0 0 0.37 User Identify & bar 40 40 -23 0.63 Nonuser -60 -60 No test -60 2 0 -11.1 0.37 User -30 Don't identify -30 -30 -11.1 0.63 Nonuser 0 0 40 C 40 40