What is the probability that the man is guilty?

Question

Problem

I try to build some connection between those text provided figure to formulate a bayes equation when I want to solve : "What is the probability that the man is guilty?" I know that: $$ P(Guilty | abuse) = P(abuse | guilty ) * P(Guilty) / P(abuse) $$ but where should those come from and what is the reason that comefrom those text. Can anyone translate those sentences to math language to ease problem?

A woman has been murdered, and her husband is accused of having committed the murder. It is known that the man abused his wife repeatedly in the past, and the prosecution argues that this is important evidence pointing towards the man’s guilt. The defense attorney says that the history of abuse is irrelevant, as only 1 in 1000 men who beat their wives end up murdering them.

Assume that the defense attorney’s 1 in 1000 figure is correct, and that half of men who murder their wives previously abused them. Also assume that 20% of murdered women were killed by their husbands, and that if a woman is murdered and the husband is not guilty, then there is only a 10% chance that the husband abused her.

Questions:

• What's prob the man is guilty?
• Is the prosecution right that the abuse is important evidence in favor of guilt?

Answer 1

In general, statements like:

• "Only $p\%$ of $X$ do $Y$."
• "$p\%$ of $X$ are $Y$"
• "If $X$ is true, then there is a $p\%$ chance that $Y$ is too."

all indicate that the probability of $Y$, given $X$, is $p\%$. That is,

$$P(Y \mid X) = p\%.$$

So, for example, the statement:

only $1$ in $1000$ men who beat their wives end up murdering them

indicates that

$$P(\text{Husband murders wife} \mid \text{Husband beats wife}) = \frac{1}{1000} = 0.001.$$

Things get a bit tricky when the language is not precise (which happens all the time in English). I opted not to use the word "abuse" because that's a far broader concept than simply beating the other party. I also decided to use "husband" instead of "man", even though it's reasonably clear from context that the men in question here are married (and married to women).

Other statements are not so clear:

Also assume that $20\%$ of murdered women were killed by their husbands

This could be interpreted a couple of ways: either it's talking about the probability of a person being murdered by a man they are married to, given that they are an adult woman (possibly married to a man, possibly not), or it's talking about the probability of a person being murdered by a man that they're married to, given that they are a woman and they are married to a man. The former probability should be smaller, since we would be considering all women, including women who never marry a man (all of whom cannot be murdered by a husband that doesn't exist). Given the context of the question, I think it's intended to be the latter; "man" and "woman" in this question generally seem to be code for husbands with wives and wives with husbands respectively.

I think, keeping this in mind, it should be possible to decode the four conditional probability statements in the question.

Answer 2

Consider the following sets of women:

1. Those who are beaten, and eventually murdered, by their husband.
2. Those who are beaten by their husband and murdered by somebody else.
3. Those who are beaten by their husband and never murdered.
4. Those who are murdered by their husband but not beaten by him.
5. Those who are murdered by someone other than their husband, and not beaten by their husband.
6. Those who are neither beaten nor murdered.

Let $P_i$ be the probabilities of a woman belonging to each of these sets.

The "$1$ in $1000$" statement says $P_1/(P_1 + P_2 + P_3) = 1/1000$. The "half" statement says $P_1/(P_1 + P_4) = 1/2$. The "$20\%$" says $(P_1 + P_4)/(P_1 + P_2 + P_4 + P_5) = 0.20$. The "$10\%$" says $P_2/(P_2 + P_5) = 0.10$. And since we've covered all logical possibilities, $P_1 + P_2 + P_3 + P_4 + P_5 + P_6 = 1$.

We'd like to know $R = P_1/(P_1 + P_2)$, i.e. the conditional probability that a woman is murdered by her husband, given that she was beaten by her husband and was murdered. We can solve for $P_1$ to $P_5$ as functions of $P_6$, and then substitute into $R$ (all of $P_1$ to $P_5$ will end up as fractions times $1-P_6$, and the $1-P_6$ factor will cancel). The result I get is

$$ R = \frac{5}{9}$$

Conclusion: the prosecution had better have some real evidence, because this does very little to help their case.