Several commenters (n+1, n+2, Trevor, math_geek, EconomistsDoItWithModels, Neil, Mark R., possibly others I’ve overlooked) solved yesterday’s probability puzzle correctly, and you can learn a lot by reading their answers. Here’s my version of their argument:
Among those who are prescribed the medicine, there are five kinds of people, represented by the five non-blacked-out squares in the following chart. A, B, C, D and E are the fractions of the population of each type.
First, we are given that 60% take the medicine, which means 40% don’t take it. That is, A+B+C = 3/5 and D+E = 2/5 . That’s two equations.
Second, we are given that if the man took the medicine, there’s a 90% chance it killed him. If he took the medicine, we know he’s in box A or B. (He’s not in C because he’s dead). So this means that 90% of the people in those two boxes are actually in box A. That is, A/(A+B) = 9/10. That’s a third equation.
Third, it is a natural, reasonable, and simple assumption that death-by-another cause is equally likely among those who take the medicine and those who don’t. That is, B/(A+B+C) = D/(D+E).
That’s a total of only four equations in five unknowns, and it does not determine any of the values A,B,C,D, or E. But rather remarkably, it does determine the ratio A/(A+B+D), which is all we care about. (We know the man is one of the boxes A, B and D because he’s dead, and we want to know what fraction of such people are in box A.)
Indeed, from these four equations it’s easy to get that A=9B and D=2B/3. From this, we have A/(A+B+D)=27/32, which is the answer.
Of course, it’s always possible that the man both had a fatal heart attack and took a fatal dose of medicine. If you unravel the argument above, you’ll see that in that case I’ve counted the man as being in box B, not box A. That’s why I added the clarifying remark that if I say “the medicine killed him”, I mean that the medicine and only the medicine killed him; in other words, “the medicine killed him” means that if he hadn’t taken the medicine, he would have lived. Apparently this remark was unclarifying for many of you, but it does seem to me that it was necessary to add this in order for the problem to be well-specified.
To put this another way, when I ask “What is the probability the medicine killed him?”, the answer depends on whether a heart-attack-plus-fatal-dose-of-meds counts as the medicine killing him. The clarifying remark was meant to remove this ambiguity.
Hat tip to Barry Nalebuff; I’ve adapted this problem from his Problem 2.