Im learning about conditional probability and although I think I finally got the hang of it, I sometimes stumble across worded problems that make me doubt whether I’m dealing with $P(A \cap B)$ or with $P(A|B)$. To illustrate my problem I’ll use an example I find particularly confusing.
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. What is the probability that the man is guilty? Is the prosecution right that the abuse is important evidence in favor of guilt?
In this example, is
only 1 in 1000 men who beat their wives end up murdering them.
the probability $P(A\cap M)=1/1000$ or $P(M|A)=1/1000$? where $A$ is the event of abusing and $M$ the event of murdering. Because I could read it as “those men who also beat their wives” which inclines me towards intersection. But I could also have read it as “given that the men beat their wives, the probability of murder is 1/1000” which sounds more like conditioning.
Similarly, I’m unsure whether
half of men who murder their wives previously abused them
is talking about $P(A|M)$ or if
20% of murdered women were killed by their husbands
is an statement about $P(M \text{ by } H| M)$, murdered by husband, $H$, given wife has been murdered. I am specially confused with
if a woman is murdered and the husband is not guilty, then there is only a 10% chance that the husband abused her.
Is this statement about $P(A|M \cap H \text{ innocent})$?
How does oneself come up with a consistent line of reasoning for distinguishing conditional probabilities from intersections?
