Problem 1

Let A_1, ..., A_n be a collection of disjoint sets such that their union is
equal to the sample space.  Prove by induction That for any event B

P(B) = P(B and A_1)+...+P(B and A_n)

Problem 2

Use problem 1 to show that if A_1,...,A_n is a disjoint collection that covers
the sample space then for any events B,C such that P(C) != 0

P(B|C) = P(B and A_1|C) + ... +P(B and A_n|C)

observe the useful corollary:

P(B|C) = P(B and A|C) + P(B and not A|C)

Problem 3

In the OJ Simpson trial it was claimed by the defence that the fact that
Simpson battered his wife should be excluded from the evidence as the fraction
of wife batteres that murder their wives is 1/2000 (Dershowitz Lemma).
In this problem we will condition this probability on the event that Nicol is 
dead and see that we obtain a somewhat different result.

We have the following statistical information:

number of women = 125M
women murdered in 1994 = 6000
women murdered by souse in 1994 = 1500
women with spouse in 1994 = 50M
women battered by spouse in 1994 = 3M

Being murdered counts as being battered. Also, the event of being murdered by
somebody else is not infuenced by whether the woman was being battered or not.

Define the events

G = Woman is murdered by spouse
M = Woman is murdered
B = Woman is battered

Use the given information to evaluate the following

a) Dershowitz Lemma: Pr(G|B) = 1/2000
b) Evaluate Pr(G|B and M)
c) Evaluate Pr(G|M)

Do you think that the evidence that Simpson battered his wife should have been
excluded based on Dershowitz Lemma?

