Two coins, one fair, one biased

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 likely1

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.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. assuming we had uniform priors to begin with. []


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