It may seem as if mathematical expectation should have been easier to grasp than probability. From an aleatory point of view the expectation is just the average pay-off in a long run of similar gambles. We can actually ‘see’ the profits or losses of a persistent gamble. We naturally translate the total into average gain and thereby ‘observe’ the expectation even more readily than the probability. However the very concept of averaging is a new one and before 1650 most people could not observe an average because they did not take averages. Certainly a gambler could notice that one strategy is in Galileo's words ‘more advantageous’ than another but there is a gap between this and the quantitative knowledge of mathematical expectation.

Cardano's notion of ‘equality’ and ‘the circuit’ in games of dice is some anticipation of mathematical expectation but it is difficult to follow in detail. Not until the correspondence between Fermat and Pascal do we find expectation well understood. This concept is at the very heart of Pascal's wager. Recall, however, that the Port Royal Logic thinks it important to ‘reorient’ people so that they base decisions on both utility and probability. This suggests that a comprehension of expectation was not something one could take for granted even in 1662. Yet shortly before there had been a really thorough statement of concepts akin to expectation. They are well worth scrutiny. I refer to the first printed textbook of probability, Christian Huygens 1657 Calculating in Games of Chance.