On unknown inequalities that may exist between supposedly equal chances

Abstract

Inequalities of this kind have an obvious influence on the results of calculations of probabilities, an influence that deserves particular attention [1]. Consider the game of heads or tails, and suppose that it is equally easy to get one or the other face when the coin is tossed. Then the probability of getting heads on the first toss is 1/2, and that of getting it twice running is 1/4. But if the coin is biased in favour of one of the faces rather than the other, and if one does not know which face is favoured by this bias, the probability of getting heads on the first toss will still be 1/2, because, if one is ignorant of the face favoured by the bias, the probability of the simple event is increased or decreased by the same amount depending on whether or no the bias is in its favour. But even in this state of ignorance the probability of getting heads twice running is increased. Indeed, this probability is found by multiplying that of getting heads on the first toss by the probability that, having thrown heads on the first toss, one will throw it on the second. Now the occurrence of heads on the first toss gives reason to believe that the coin is biased in favour of this side. Consequently in such a case the unknown bias increases the probability of getting heads on the second toss, and as a result it increases the product of these two probabilities. As a numerical example, let us suppose that this bias increases the probability of the favoured simple event by a twentieth.