Al and I talked about the final together. we each took about 1 hour 20
min to complete it. Here are my comments, including stuff Al and I
talked about:

1. third question: I guess our graph is connected, yes? I admit I don't
   know how to solve this one, though.
   9th question: what's the answer? I can think of a surjection from
     [0,1] to [0,1]^2 (alternate digits: .2414256... maps to 
     (.2126...,.445...) so it seems plausible to me that a surjection
     from R to R^2 exists (since there's a bijective map from [0,1]
     to R). Am I mistaken, or are the students supposed to have a similar
     line of reasoning?
   final question: Are we trying to make this one this easy? it seems obviously
   false to me. or did we want to say, n!=2^{f(n)}

2. we haven't done many examples like this... actually, I think the only
   example we did this term showed up in the quiz review. is a=32 the minimal
   $a$ we want?

3d. people throw dice, not flip them (sorry to be picky)

4b. technically, it's not the LHS and RHS of the identity that are the number
   of ways... we need to multiply by $r$ on both sides first, and we should
   change either the equation in 4a or the problem statement in 4b to
   reflect that.

6. there's a fair amount of computation involved here. it'll make things
   easier on the students (and us as graders) if we tell them 8!=40320.

7. it'll be easier to grade this question for partial credit if we name
   the events, e.g., H is the event of having high blood pressure, and P
   is the event of testing positive

9. Al and I don't understand what this problem is asking. If a try is
   a "sequence of pulled balls," how does a try "consist of one or two
   consecutive balls"? (We haven't been able to solve the three parts
   of the problem, either.)

10. the method of repeated diagnoses will only work if we assume some level
    of independence of diagnoses.

12b. Al notes that 12b yields a tighter bound than part a. it's strange
     but okay. **oops** we used the var/(var+x^2) formula but forgot 
     to square x.

12c. if we're going to use standard dev on a problem, we should include
     the formula standard dev=\sqrt{variance}.