ME290M: Expert Systems

Due Date: Thursday, March 18, 1999

1. Other Logics (10 points)

Give five examples of facts from the mechanical engineering (or your own discipline) domain that are difficult to represent and manipulate in classical predicate logic. Express them in one of the following representations (use the one you consider most appropriate for the problem): fuzzy logic, default reasoning or probabilistic logic. If none of these representations are appropriate, what do you recommend?

2. Solve Problem 4.1 (a) in the text. (10 points)

Pr(A+B) = Pr(A) + Pr(B) — Pr(A+B)

Hint: Any event X+Y can be written in the form X+(X'Y) where X and X'Y are mutually exclusive. Also, write B as the union of two mutually exclusive sets.

3. Solve Problem 4.3 in the text. (10 points)

Three bins contain some working and some defective components as follows:

1. Create a probability table showing outcomes and prior, conditional, joint, and posterior probabilities.
2. What is the probability of each drive crashing within one year?
3. What is the probability of any drive crashing within one year?
4. For a crash (on Fred's drive) within one year, what is the probability it is Brand X, Y or Z?

5. Solve Problem 4.5 in the text. (10 points)

(a) Make a table showing the outcomes, and prior, conditional, joint, and posterior probabilities.

(b)What percentage of people who test positive don't have the disease (false positive)?

(c) What percentage of people who test negative do have the disease (false negative)?

6. Solve Problem 4.6 in the text. (10 points)

To show that pairwise independence does not necessarily mean mutual independence, define the following events for the toss of two dice.

A = even first die

B = even second die

C = even sum

7. Solve Problem 4.7 in the text.(10 points)

For mutually exclusive sets A and B

8. Solve Problem 4.9 in the text. (10 points)

A disk drive may malfunction with either fault F1 or F2, but not both. The possible symptoms are:

A = Event that there are both {bad writes, bad reads}.

B = Event of {bad reads}

And F2 is three times as likely to occur as F1.

For this type of drive:

Pr(A|F1)=0.4

Pr(A|F2)=0.2

Pr(B|F1)=0.6

Pr(B|F2)=0.8

What is Pr(F1|A)?, Pr(F1|B)? Pr(F2|A)?, Pr(F2|B)? You may assume the symptoms (A, B or C) can only happen if a failure occurs (i.e., if F1 or F2 occurs).

Extra Credit: Define C = Event of (bad writes}. What is Pr(F1|C) and Pr(F2|C)?

9. The Curious Competitor (20 points)

Of three finalists in a design competition, named A, B, and C, one will be announced the official winner the next day. The panel has met and the Judge has made his final decision. Competitor A assigns a probability of 1/3 to the event that any one of the three competitors will win. Being curious, Competitor A tries to get some information out of the Judge without asking directly. He knows that regardless of who wins, at least one of the other two competitors will be losers (both if A is the winner). He asks the Judge to give him the name of one of the losers. The Judge, however, refuses to grant the request, arguing that if A knew which of the other two was a loser, the probability that A would be the winner would increase to 1/2 instead of 1/3. Do you agree with the Judge's argument? If so use conditional probability to verify his statement. If not use the axioms and theorems of probability and event algebra to find the probability that A is the winner given that the Judge names B or C as one of the losers. Assume that if both B and C are losers, the Judge would be equally likely to pick either one.

Hint: In problems of this kind, it is important to define precisely the events. It may be helpful to define the following events:

A = Competitor A wins

B = Competitor B wins

C = Competitor C wins

Ja = Judge says B loses

Jb = Judge says C loses