This was invented by David McKay, but I learned of it via Allen B. Downey.

We suspect Oliver of committing murder. Tests at the crime scene reveal two different blood types, O and AB, are present in stains that could not have come from the victim. Alas, we don’t have access to DNA tests in this hypothetical, but we do know Oliver has type O blood. We also know that roughly 60% of the public share Oliver’s type, yet a mere 1% of people have AB blood.

Question: how does this evidence effect the odds of Oliver being the perpetrator?

The best approach to solving this is Bayes’ Theorem in Odds Ratio form. Let’s start by assuming Oliver’s guilty. That means the odds of him being the O donor is 100%. We need an AB donor, too, so they contribute 1% toward the total. The odds for that hypothesis are thus (100%) * (1%) = 1%.

Next, we assume Oliver is innocent. That means we need to draw two people from the general population; let’s call them X and Y. If X happens to have O-type blood, which carries a 60% chance, then Y must have AB-type, which is 1% of that 60%. If X doesn’t have O-type, there’s still a 1% chance they could be the AB donor. Y must then be O-type, with has a 60% chance of happening. So the odds of Oliver’s innocence are (60%)*(1%) + (1%)*(60%) = 1.2%

So the Bayes’ Factor for Oliver’s guilt versus his innocence is 0.01 / 0.012, or 5/6 or .83333 if you insist on decimals. Whatever the case, that means the blood type evidence argues against Oliver being the perpetrator.

I know, you’re probably thinking I did something fishy when calculating Oliver’s innocence. Why does order matter when we’re selecting from the public? It’s much like I laid out above: the first person you pluck out can satisfy the blood evidence in two separate ways, which means we need to branch down two paths. Hence, we need to count twice.

You’re probably still sketchy on the whole thing. That’s cool, because I have something you can’t argue against: brute-force numeric simulation.

    Trials	H1 success	H2 success	   BF for H1/H2
   5000000	     49992	     59896	       0.834647

Statistics can be a tricky beast, can’t it?