Probability Generating Functions — The Casino Owner’s Best Friends
Probability Generating Functions are extremely powerful ways of understanding how game rules can be safely adapted
Let’s imagine that you own a Casino, and one of the most popular games there is Craps. Craps involves the throw of two normal dice, and its all about the total score from the dice — which can range from 2 to 12. You are looking to spice things up for your customers and offer them some fun variations on their favourite game.
One variation you are considering is offering them to choose alternative ‘biased’ dice to play with. For example, maybe they can choose a die that has the numbers 1, 2, 2, 4, 5, 6 along with a regular 1, 2, 3, 4, 5, 6 die. The possible score range is still 2 to 12, but maybe the probabilities of a given total score have changed, and as a Casino owner you probably would want to know that to make sure that you aren’t going to be out of pocket on the odds from a specific choice of dice.
Let me introduce you to a powerful tool to solve problems like this — the probability generating function.
