d'Alembert's Misstep

Such is the character of the doctrine of chances that simple-looking problems can deceive even the sharpest minds [Gorroochurn, p. 191]. Below I want to discuss a misstep committed by Jean le Rond d'Alembert (1717-1783) a foremost intellectual of the eighteenth century, co-editor with Denis Diderot of the Encyclopédie.

In his article Croix ou Pile in the Encyclopédie he considered the following problem:

In two tosses of a fair coin, what is the probability that heads will appear at least once?

The sample space of two coin tosses is $\{HH,HT, TH, TT\},$ with each of the elementary events judged to be equally probable such that, since heads appear in three of them out of four, the answer to the question is $\displaystyle\frac{3}{4}.$ However, d'Alembert's answer was different: $\displaystyle\frac{2}{3}.$ He reasoned that in real life no one would continue the experiment after heads showed up on the first toss. In other words, d'Alembert's sample space was shortened to $\{H, TH, TT\},$ prompting the answer he gave. By doing this d'Alembert was proposing a model, according to which equal probabilities are assigned to observable events.

What is wrong with this model?

The idea of equiprobability stems from the abstract concept of symmetry: in the absence of other information, equipossible events should be considered as equiprobable. Obviously, in the case of the coin tossing experiment, d'Alembert had a notion of which events are equipossible different from the accepted norm. His view led to an erroneous solution to the problem.

I propose to consider the problem of equiprobability in the framework of a nuanced symmetry principle:

The probability assigned to an event should be independent of the sample space in which the event may occur.

In this generality, the principle is not quite true. For example, the probabilities of having heads in two tosses is different from having heads in three tosses. But these events should not be considered the same because, by their nature, they belong to different sample spaces. But say, having heads on the first toss is an event that may occur in the sample spaces of two and also of three tosses. In the former it's the event $\{HH, HT\},$ in the latter it's $\{HHH, HHT, HTH, HTT\}.$ In both cases, if the elementary events are judged equiprobable, the probability of heads on the first toss comes out the same: $\displaystyle\frac{1}{2}=\frac{2}{4}=\frac{4}{8}.$

In d'Alembert case, the event $\{H\}$ is the same as $\{HH,HT\}$ in two tosses and $\{HHH, HHT, HTH, HTT\}$ in three, and thus should be assigned the probability $\displaystyle\frac{1}{2}$ whereas the events $TH$ and $TT$ each have the probability of $\displaystyle\frac{1}{4}.$

d'Alembert was wrong because in his model to answer the question of the probability of heads in three throws, he would consider $\{H, TH, TTH, TTT\}$ as the sample space and assign all four events equal probabilities, making the probability of the same event $H$ $\displaystyle\frac{1}{4}$ different from $\displaystyle\frac{1}{3},$ as in two tosses.

So, I think, one facet of the Symmetry Principle is that the probability of an event should be independent of the sample space in so far as the nature of the event leaves us a choice of sample spaces.

Reference

  1. P. Gorroochurn, Errors of Probability in Historical Context, in The Best Writing on Mathematics 2013 (M. Pitici, Editor), Princeton University Press, 2014
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