There were many excellent comments on yesterday’s Bayesian Riddle. Here’s what I believe is the simplest and most natural analysis.
First, let’s recall the problem:
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?
Notice that if you considered all the suspects equally likely, your estimate would have been three to one for a smoker. Since you estimated only 2-to-1, you must have believed that the individual smokers were less likely than average to be guilty. So when you find out the culprit is female, it’s the female non-smoker — that is, Alice — who is now your prime suspect.
To quantify this, 2/3 of the original probability weight is on the three smokers, which means they each have a 2/9 chance of being guilty. 1/3 of the weight is on Alice, the non-smoker. A ratio of 1/3 to 2/9 is the same as a ratio of 3 to 2. So when you narrow things down to one smoker and one non-smoker, the odds are 3 to 2 for the non-smoker. That means there’s a 60% chance your culprit is Alice.
It is, of course, in the nature of these puzzles that one can get pretty much any answer one wants by introducing unnatural assumptions. If you’d already excluded Bob before you came up with your 2-to-1 estimate, then the others all have equal probability weight, so after Ted is excluded, the odds for Alice are 50/50. For that matter, maybe you’d already excluded Carol, in which case, after the investigators rule out the boys, the odds for Alice are 100%. Or maybe the investigators concluded the killer was a woman because they knew the killer was Carol. (Or, as one commenter cleverly suggested, maybe they concluded the killer was a woman because they found a cigarette with red lipstick, which causes you to update the odds that the killer was a smoker.) Et cetera. But all of these stories assume that I withheld critical information about how you and your investigators reached your conclusions. So unless you suspect me of playing unfair, the answer is Alice with a probability of 60%.