Did a gambling doctor write the first book on the odds of dice?

Yes, and its author was a gambler. Around 1564 the Italian doctor Gerolamo Cardano wrote Liber de ludo aleae, the Book on Games of Chance, which contains the first systematic treatment of probability. It also contains a section on effective cheating methods. It sat unpublished until 1663.

Cardano was a polymath: mathematician, physician, astrologer, philosopher, and gambler, all at once. He was chronically short of money and kept himself solvent by being an accomplished gambler and chess player, so the gaming table was both his income and his laboratory. He used the game of throwing dice to understand the basic concepts of probability.

His central move was simple. He showed that you can define odds as the ratio of favourable to unfavourable outcomes. That implies that the probability of an event is the number of favourable outcomes divided by the total number of possible outcomes. A single die is the easiest case: it lands showing a whole number from one to six, each value equally likely. Add two dice together and the picture changes, because some sums, such as 7, are more likely than others, such as 2 or 12.

The treatment was still a first draft. Cardano was aware of the multiplication rule for independent events, but he was not certain about what values should be multiplied. Aside from his elementary work, the doctrine of probabilities dates to the letters between Fermat and Pascal in 1654, and Christiaan Huygens gave the earliest known scientific treatment in 1657.