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Classic Problems of Probability by Prakash Gorroochurn

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Problem 20

Three Coins and A Puzzle from Galton (1894)

Problem. Three fair coins are tossed. What is the probability of all three coins turning up alike?

Solution. The sample space Ω = {HHH, HHT, . . ., TTT} is made up of eight equally likely outcomes. Of these, all three coins turn up alike with either HHH or TTT. The required probability is therefore 2/8 = 1/4.

20.1 Discussion

Sir Francis Galton (1822–1911) is considered to be the father of eugenics and one of the most renowned statisticians of his time (Fig. 20.1). His name is usually associated with the notions of correlation and the regression effect.1 In the February 15, 1894 issue of Nature, Galton presented both the correct solution and an intentionally wrong solution to the above problem (Galton, 1894). Concerning the wrong solution, he writes

I lately heard it urged, in perfect good faith, that as at least two of the coins must turn up alike, and it is an even chance whether a third coin is heads or tails; therefore the chance of being all-alike is as 1 to 2, and not as 1 to 4. Where does the fallacy lie?

Figure 20.1 Sir Francis Galton (1822–1911).

img

The argument does seem to contain an infallible logic. The fallacy, however, lies in the assumption that, given that at least two coins turn up alike, then there is an even chance for the remaining coin to be alike too. We can show this as follows. To fix ideas, let us consider ...

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