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Classic Problems of Probability by Prakash Gorroochurn

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Problem 13

D'Alembert and the Gambler's Fallacy (1761)

Problem. When a fair coin is tossed, given that heads have occurred three times in a row, what is the probability that the next toss is a tail?

Solution. Since the coin is fair, the probability of tails (or heads) on one toss is 1/2. Because of independence, this probability stays at 1/2, irrespective of the results of previous tosses.

13.1 Discussion

When presented with the problem, d'Alembert insisted that the probability of a tail must “obviously” be greater than 1/2, thus rejecting the concept of independence between the tosses. The claim was made in d'Alembert's Opuscule Mathématiques (Vol. 2) (d'Alembert, 1761, pp. 13–14) (see Fig. 13.1). In his own words

Let's look at other examples which I promised in the previous Article, which show the lack of exactitude in the ordinary calculus of probabilities.

In this calculus, by combining all possible events, we make two assumptions which can, it seems to me, be contested.

The first of these assumptions is that, if an event has occurred several times successively, for example, if in the game of heads and tails, heads has occurred three times in a row, it is equally likely that head or tail will occur on the fourth time? However I ask if this assumption is really true, & if the number of times that heads has already successively occurred by the hypothesis, does not make it more likely the occurrence of tails on the fourth time? Because after all it is not possible, it is even ...

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