The following excerpt is from Unweaving the Rainbow (1998) by Richard Dawkins.

A recent highly publicized case in America, where the jury were systematically confused about DNA evidence, has also become notorious for another piece of bungled probability theory. The defendant, who was known to have beaten his wife, was on trial for finally murdering her. One of the high-profile defence team, a Harvard professor of law, advanced the following argument: Statistics show that of men who beat their wives, only one in 1000 go on to kill them. The inference that any jury might be expected to draw (indeed, were intended to draw) is that the defendant’s
beating of his wife should be discounted in the murder trial. Doesn’t the evidence show overwhelmingly that a wife-beater is unlikely to turn into a wife murderer?

Do the following in order to expose this misleading defence argument.
(a) Draw a Venn diagram with S defined as the set of all wives, B as the subset of wives that are beaten by their husbands, M as the subset of wives that are murdered, and H as the subset of wives that are murdered by their husbands.
(b) Using the symbols defined above, write down the conditional probability that the defence counsel put forward.
(c) Since we know that someone has murdered his wife, write down the pertinent conditional probability that should have been used.
(d) Provide an algebraic proof that the probability in (c) is greater than or equal to that in (b). Hint: Look at the Venn diagram from (a). (How much more incriminating might this larger probability have been?)

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This is what I have so far, but I'm not sure if I'm going in the right direction.

(a)

(b) P(H|B) = P(H intersect B)/P(H)

(c) P(H|M) = P(H intersect M)/P(H)

(d) algebraic proof...a bit lost here

Your Venn diagram is wrong, it should look like that is the attachment.

RonL

Attached Thumbnails

 

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

oh i see!
i wonder if that changes my probabilities?

anyhow, thanks for your help! i'm going to work on the problem some more now.

Yes I think you should. Do you have Dawkins’ book? Or is that just a quote someone gave you? I looked up the quote, I read the book 10 years ago and had forgotten the details. Dawkins points out that one I.J. Good discussed this in Nature, 375, 541: The title is “When a batterer turns a murderer”.

I think that you should have been given Dawkins’ entire discussion if this is going to be a meaningful probability exercise.

No, it's actually a quote that was given to me by my prof.
Now I actually want to read up on his entire discussion. lol!