See instructions on the assignments page on how and when to turn these in. Be sure to show your work and justify all of your answers.

1. Exercise 7.9 in R&N.

2. Exercise 7.12 in R&N.

3. Exercise 13.5 in R&N.

4. There are three condemned prisoners A, B and C.  The governor has announced that one of the three, chosen at random, has been pardoned, but doesn't say which.  Prisoner A, realizing that he only has a 1/3 chance of having been pardoned and anxious to learn his fate, reasons with the warden as follows: "Please tell me the name of one of the other prisoners B or C who will be executed.  I already know that at least one of them will be executed so you will not be divulging any information."  The warden thinks for a minute and then asks how he should choose between B or C in case both are to be executed.  "In that case," A tells him, "simply flip a coin (when I'm not around) to choose randomly between the two."  Reluctantly, the warden agrees and the next day tells A that B will be executed.  On hearing this news, A smiles and thinks to himself, "What a fool this warden is to fall for my trick.  Now my chances of having been pardoned have increased from 1/3 to 1/2"  (since either A or C must have been pardoned).

1. Was A correct in his final assertion?  Use Bayes rule to compute the probability that A was pardoned given that the guard tells him that B will be executed.  (Note that this is not the same as the probability that A was pardoned given that B will be executed (i.e., was not pardoned).  This is because the event that "the guard tells A that B was not pardoned" is different from the event that "B was not pardoned".  This is a tricky problem so be careful to distinguish these two events.)
2. Suppose, in the event that the warden has to choose between B or C, that he chooses B with probability p (rather than 1/2 as above).  Now what is the probability that A was pardoned (in terms of p) given, as before, that the guard tells him that B will be executed?  (If you wish, you may answer part b first, and then answer part a by setting p=1/2.)

5. Exercise 14.2 in R&N.

6. Consider the burglar alarm example in R&N Figure 14.2 (and discussed extensively in class).  In R&N (page 505), it is shown that the probability of a burglary, given that both John and Mary call, is roughly 28.4%.  Suppose now that you hear on the radio that there was an earthquake in your area.  Now what is the probability of a burglary (given that JohnCalls, MaryCalls and Earthquake are all equal to true)?  Has the probability of a burglary increased or decreased as a result of this new information?  Intuitively, why does this make sense?  (This phenomenon is called "explaining away".)

7. Exercise 14.3 in R&N.

8. Exercise 14.4 in R&N.

9. Exercise 14.7 in R&N.

10. Consider two discrete variables x and y each having three possible states, for example x,y ∈ {0,1,2}. Construct a joint distribution p(x,y) over these variables having the property that the value xˆ that maximizes the marginal p(x), along with the value yˆ that maximizes the marginal p(y), together have probability zero under the join distribution, so that p(xˆ,yˆ)=0.