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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?
Yiffany
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    Please [edit] the question to show us what you have tried and where you are stuck. Possibly helpful, possibly duplicate: https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 – Ethan Bolker Aug 07 '23 at 00:06
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    I would very much encourage you to change the setting of your problem. The setting you have provided does not change the underlying mathematics in the slightest, and is likely to be offensive to a fair number of people. – Xander Henderson Aug 07 '23 at 00:07
  • @XanderHenderson, OK changed, thanks – Yiffany Aug 07 '23 at 00:17
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    @XanderHenderson The suggestion in your comment is well taken but problematic. There are subtleties that might well be lost in translation. For example, assuming that 1 in 1000 abusers murder the abuse victim, what is the probability that the Person-A abuses Person-B, who is then murdered by someone else. This relevant statistic is generally glossed over by defense attorneys in the business of selling soap to the jury. – user2661923 Aug 07 '23 at 00:18
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    I don't think this is offensive. Crime exists in the world. Nothing is gained by pretending it doesnt. – Rohit Pandey Aug 07 '23 at 00:32
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    @RohitPandey I did not say that everyone would find it offensive. I did not even say that a majority of people would find it offensive. I said that a "fair number" of people are likely to find it offensive. The fact that you are not offended is not really relevent. – Xander Henderson Aug 07 '23 at 01:43
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    @user2661923 If those kinds of things are important to this question, then this is no longer a question about mathematics, but a question about forensics. This is a mathematics Q&A site. It should be possible to present a problem with a framing that is not likely to offend (or it should be possible for the asker to explain why this particular framing is crucial to the problem by, for example, citing a source for the problem, or explaining how it came up). – Xander Henderson Aug 07 '23 at 01:46
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    @XanderHenderson I agree: ideally this should be about something else. However, the source of confusion appears to be the process of extracting the mathematical content from the text. The asker can't really be expected to change the subject, while keeping the mathematical content the same, if they can't read the mathematical content. Someone else could attempt to rewrite it as an "isomorphic" word problem, but we'd run the risk of accidentally rewriting out the source of confusion. If anyone wants to try this, I'd be happy to adjust my answer. – Theo Bendit Aug 07 '23 at 02:06
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    @TheoBendit Indeed, which is why I included the parenthetical "or it should be possible for the asker to explain why this particular framing is crucial to the problem by, for example, citing a source for the problem, or explaining how it came up". – Xander Henderson Aug 07 '23 at 02:10
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    @XanderHenderson - on any given thing, there is some proportion of people that will get offended. What is a "fair amount" and what are the thresholds for this "fair amount"? Who decides those thresholds? Changing the nature of a question like this crosses the line. On the surface, you could create n scenarios that keep the math close. But digging in deeper, qualitative differences are bound to emerge. – Rohit Pandey Aug 07 '23 at 04:00

2 Answers2

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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.

Robert Israel
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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.

Theo Bendit
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  • "Things get a bit tricky when the language is not precise.", this might need me add the second part of this question, they ask "is the prosecution right that the abuse is important evidence in favor of guilt?" It seems skeptical about that 1/1000? – Yiffany Aug 07 '23 at 02:32
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    @Yiffany Ah, I see what you mean. I would agree that the answer, given the assumptions in the question, should be $1/1000$. That's the probability that this man, given all the evidence we've been presented (i.e. he beat his wife), has killed his wife. Perhaps the author wanted you to find the probability $P(\text{Guilty})$, i.e. the probability that any husband has murdered his wife? But this number would not practically help in the court case, as it would apply to every husband of every wife, regardless of information about abuse. I'm not really sure what the point of the question is. – Theo Bendit Aug 07 '23 at 02:45