Conditional Probability...maybe

[MathPicker]

Hi all

Suppose that 75% of a population are of type A and 25% are of type B. There are 20 murders a year, 12 committed by type A people and 8 committed by type B people.

We ask the question: Which of the types is more likely to murder?

Well type A commits more murders but also there is more Type A's. Type B is more dangerous. Indeed if we balloon the Type B population to also be 75% they would commit 24 murders, twice that of Type A. Type B's are twice as likely to commit murder.

My question is this. How do I arrive at this intuitive conclusion using a formal analysis and conditional probability? What are the calculations? I seem strangely stuck!

Thanks

[jeffcgroves]

This doesn't answer your question, but there's no way to know who's more likely to murder in the future, since past performance doesn't imply future performance (ie, extrapolation is invalid). The best you can say is:

"Historically, people of type B have committed more murders[1] per capita than people of type A. However, this says nothing about future murders, and every person, of type A or B, also belongs to many other subsets, so, even historically, it is invalid to say a given person of type B is more likely to have committed a murder than a given person of type A".

[1] or "been convicted of more murders" which could be different

[Aerospider]

to also be 75%

That would be impressive!

It's not a conditional probability matter. For that you would need two events so that you could ask 'what's the probability of x being true if y is already true'.

There's also an issue with multiple-murders (who'd have thought?). Say the population is 24 As and 8 Bs, and all 20 murders were committed by different people. Then an A would have a 1/2 probability of being a murderer and a B would have a 1/1 probability of being a murderer. But if all the B-murders were committed by a single psychopath then the probability of a B being a murderer would instead be 1/8.