# Homework Assignment #3

#### Due on paper in class, Wednesday 23 April 2008

There are twelve questions for 100 total points. credit. All but the last one are from the textbook, Mathematical Foundation for Computer Science. (The last two are not yet listed.) Note that the book has both Exercises and Problems -- make sure you are doing a Problem and not the Exercise with the same number. Numbers in parentheses following each problem are its individual point value.

Problem X.3 added 13 April 2008.

• Problem P.5.1 (5)

• Problem P.5.2 (5)

• Problem P.5.5 (5)

• Problem P.6.1 (10)

• Problem P.6.2 (5)

• Problem P.7.2 (10)

• Problem P.7.3 (10)

• Problem P.8.3 (10)

• Problem P.8.5 (10)

• Problem X3 (20) The police have arrested a suspect in a crime, and they have a witness who may be able to identify her. The witness is 80% accurate, meaning that the guilty person will appear guilty to him with probability 0.8, and any innocent person will appear guilty with probability 0.2 -- the witness' errors on different people are assumed to be independent. The police plan to present the witness with a lineup consisting of their suspect and four decoys whom they know to be innocent. Plan A (familiar to viewers of TV shows) is to show the witness all five people at once. If the witness sees exactly one guilty person, he will identify her -- if he sees zero or more than one, he will not identify anyone (and won't say which is the case). In Plan B, the police will show the witness the suspect and four decoys in a random order, one by one. If the witness sees a person who appears guilty to him, he identifies her and the process stops (even if there are more people to see).

There are thus three possible outcomes to each process:

• The witness identifies the suspect.
• The witness identifies a decoy.
• The witness does not identify anyone.

Your first job is to determine the probability of these three events in the case when the suspect is guilty, and again in the case when the suspect is innocent. You are doing this for both Plan A and Plan B.

Then compute, for each of the three outcomes and both Plan A and Plan B, the posterior probability that the suspect is guilty given a prior probability of 50% and then again for a prior probability of 90%.