When the redoubtable @cuttheknotmath (Alexander Bogomolny) poses the following question:

… you know there must be Something Up. Surely (the naive reader thinks) the one with two heads out of three is the one with a probability of two heads in three? But equally surely (thinks the reader who knows Alexander doesn’t set trivial problems), it can’t be that simple.

### So let’s work it out.

We have two coins, label them A and B. Coin A has given us one head from a single throw; coin B has given us two heads from three tosses.

Suppose, first of all, that coin A is the fair one. The probability of the given result under these circumstances is $\frac{1}{2} \times 3\br{\frac{2}{3}}^2 \frac{1}{3} = \frac{2}{9}$.

Suppose instead that coin A is biased. In this case, the probability is $\frac{2}{3} \times 3\br{\frac{1}{2}}^3 =\frac{1}{4}$.

We were right to be suspicious! A quarter is greater than $\frac{2}{9}$, so the second explanation is more likely ((assuming we had uniform priors to begin with.))

Applying Bayes’ Theorem, we’d assess a probability of $\frac{9}{17} \approx 0.5294$ of coin A being biased.

### So is there any intuition behind this, or is it just A Thing?

Probability is always tricky to find an intuition for (for me, at least). The best explanation I can come up with is that two-in-three heads is not that much more likely with a biased coin than a fair one (for a fair coin, it’s the equal-most likely outcome, coming up $\frac{3}{8}=0.375$ of the time; the biased coin comes up $\frac{4}{9} \approx 0.444$ of the time.)

Proportionally (as well as numerically), that’s a smaller difference than between the one-half and two-thirds.

### The moral of the story

I’m all for intuition. I use mine all the time. However, reasonable intuition lets one down here - the obvious solution is the wrong one.

That said, if you develop an intuition that says “is there a reason to doubt the obvious answer?”, your mathematical skills will be all the better for it.

* Edit 2018-07-09: It’s a complete coincidence that this post came out within a few days of Alexander’s untimely passing. I’ll miss him and his thought-provoking problems greatly.