# Suspicious coins

I quote, without attribution (unless you want me to):

My friend Bengt says that on the first day he got the following sequence of Heads and Tails when tossing a coin:  H H H H H H H H H H. And on the second day he says that he got the following sequence:  H T T H H T T H T H.

Which day-report makes you suspicious?

Most people I ask this question says the first day-report looks suspicious. But actually,​ both days are equally probable! Every time you toss a (fair) coin there is the same probability (50 %) of getting H or T. Both days Ben makes equally many tosses and every sequence is equally probable!

While the above is true in the sense that if the coin is fair then both sequences are equally probable, but the question remains: which report makes you suspicious?

As so often, there seems to be an implied assumption here that seems only common sense:

That one only has more reason to be suspicious about something when it is less probable.

But from a mathematical perspective a probability is just a measure against some standard. In which case we are not told originally told that the coin is fair but have simply assumed it and then calculated the probabilities. So it seems our ‘reasons to be suspicious’ depend on our prior expectations or assumptions. Thus, for example, if, in the run-up to the financial crash of 2008 you had thought such a crash impossible, then nothing could make you suspicious that there might ever be a crash. This may be rational (in a strange academic sense) but hardly sensible.

Lets try to apply some straightforward Bayesian probability theory, just to see how it applies. For simplicity I suppose that Bengt was tossing different coins on each day. (The other case uses the same maths and reaches the same conclusion, but with different detail.)

If we regard the example as an academic puzzle then by long-standing convention in such puzzles,  the coin is supposed to be fair. In this case the likelihood of any sequence of length 10 is 0.5^10. So far, I agree. Obviously, in such a context we ought not be suspicious of the question: they are never tricks.

On the other hand, if we imagine that there was a real Bengt, the situation is different. Let P(Head)=p, which may be 0.5 or not. Then the likelihood for the first sequence is, p^10, whereas for the second sequence it is (p*(1-p))^5. When p>0.5 the first likelihood is > (0.5)^10, whereas the second is < (0.5)^10.

Thus for the first sequence the generalised likelihood (strongly) suggests that the coin is unfair, whereas the second sequence (weakly) suggest that it is fair. To put it another way, according to Bayes’ rule, your subjective probability that the coin is fair ought to be diminished when you see the first sequence, increased (slightly) by the second.

I have never seen an actual mathematical theory of ‘suspicion’, but it seems to me that you are entitled to be suspicious when the evidence makes your assumptions seem less likely. That is:

Surely our supicions should be of our prior expectations and assumptions, rather than being straight-jacketed by them?

In the example, the account is ‘as if’ we could only ever be suspicious that the sequence might have arisen by chance, whereas I am taking the opportunity to raise suspicions about all the hidden assumptions and beliefs. (Much like Keynes’ Treatise.)

Dave Marsay

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