Economist Steven Landsburg recently posed the following Bayesian riddle here.
A murder has been committed. The suspects are:
- Bob, a male smoker.
- Carol, a female smoker.
- Ted, another male smoker.
- Alice, a female non-smoker.
You are quite sure that one (and only one) of these suspects is the culprit. Moreover, after carefully examining the evidence, you’ve concluded that the odds are 2-to-1 that the culprit is a smoker.
Now your crack investigative team, in which you have total confidence, reports that, on the basis of new evidence, they’ve determined that the culprit is definitely female.
Who’s the most likely culprit, and with what probability?
Here is his solution.
prior probability 
I haven’t looked at the solution but wouldn’t those clues narrow it down to Carol? I am not sure what that 2-1 odds clue means
That’s what I thought at first, but it’s actually Alice. Here’s why (according to Landsburg): step1: there is a 2/3 prior probability that the three smokers are guilty of the crime (because the prior odds of the culprit being a smoker are “2 to 1”), and this in turn means that each smoker has a 2/9 chance of being guilty. Thus, step 2, there is a 1/3 prior probability that Alice, the non-smoker, is guilty. Step 3: since a ratio of 1/3 to 2/9 is the same as a ratio of 3 to 2, when you narrow your inquiry down to 1 smoker and 1 non-smoker, the odds are now 3 to 2 for the non-smoker, and that in turn means there’s a 60% chance the culprit is Alice.
Thanks for the explanation Enrique. I will have to go over it a few times to get a full understanding.