The probability that the man is guilty

bayesianprobabilityprobability theoryself-learning

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?

Best Answer

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.

Related Question