Homework Set 3, Due Thurs. Oct. 2 in Class
(late papers OK until 1:00 Friday)

1. [Blood Test] A large number, N, of people are subjected to a blood test, the result of which is either "positive" or "negative". It can be given in two ways:
   i). Each person can be tested separately, so N tests are required.
   ii). The blood samples of k persons 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 required for the k people.
   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 X of tests necessary under plan ii).? [Assume that N is divisible by k]
   c). For small p, show that the value of k that will minimize the expected number of tests made under plan ii). is approximately 1/sqrt(p).
   [Note: for x close to zero we have (1 - x)^k is approximately   1 - kx.]

2. (The 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?
   a). Do a computer simulation of at least 150 trials of this. In one case you always switch, in the other you never switch.
   b). Give a probabilistic analysis. A "tree diagram" may be useful.

3. In another oft-cited experiment, psychologists 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.

4. 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?

5. (Triangles)   If you break a stick at random into three pieces, what is the probability that you'll be able to assemble the pieces into a triangle? For instance if you break a 20 inch stick into pieces of length 12.1, 4.7, and 3.2 inches, you won't be able to assemble them into a triangle (why?).
   Give both a theoretical probabilistic analysis and do a computer simulation using a random number generator.
   NOTE: There are at least three similar ways to state this problem. They give completely different results.
   a). You make two independent random cuts on the stick.
   b). You make one cut on the stick and then make the second cut on the left-portion of the stick.
   c). You make one cut on the stick and then make the second cut on the longer piece.
   This problem is asking for a) only.
   You may also find it interesting to do b) and c) for extra credit.

6. Use the Monte Carlo method for estimating the area of the ellipse
   x²/9 + y²/16 < 1. This ellipse lies inside the box
   Q = { (x,y) : |x| < 3, |y| < 4 }. Remark: The Monte Carlo method says to pick lots of points at random in the box Q. The area of the ellipse will correspond to the percentage of the points that in the ellipse. [Use at least 10,000 random points].
   Warning. In perl one writes x² as x**2 [not as x^2].
   [Remark: As a check, the area of the general ellipse   x²/a² + y²/b² < 1   is   abπ. When a=b this agrees with the usual formula for the area of a disk of radius a, so for our case Area = 12π = 37.7...].

7. Say a 20 year old friend is tested for a relatively rare cancer that occurs in only 1 out of every 1,000 people her age. The test is 99% accurate in the sense that only 1% of those who do not have the cancer still test positive and 95% of those who have the cancer test positive.
   a). If your friend tests positive, what is the likelihood that she has the cancer?
   b). If your friend tests negative, what is the likelihood that she has the cancer?
   c). Repeat parts a) and b) if only 1 out of every 10,000 people her age have the cancer.

8. Your next-door neighbor has a rather old and temperamental burglar alarm. If someone breaks into his house, the probability of the alarm sounding is .95. In the last two years, though, the alarm has gone off on five different nights, each time for no apparent reason. Police records show that the chance of a home in your neighborhood being burglarized on any given night are 2 in 10,000.
   If your neighbor's alarm goes off tonight, what is the likelihood his house is being burglarized?

9. During a power blackout, 100 people are arrested on suspicion of looting. Each is given a polygraph test. From past experience it is known that the polygraph is 90% reliable when administered to a guilty suspect and 98% reliable when given to someone who is innocent.
   Suppose that of the 100 suspects, only 12 were actually involved in any wrongdoing. What is the probability that a given suspect is innocent given that the polygraph says she is guilty?