Playing Cards with Bayes' Theorem
Wizard 330 is a home-brewed variant of Wizard, a trick-taking card game. (Think Spades or Euchre. That sort of game.) While strategizing, an interesting problem arises that can be solved very cleanly with some probability. To understand the problem, you don't need to understand the (admittedly very complicated) details of the game. All you need to know is the following: Wizard 330 is played with a deck of 90 cards whose contents are unknown at the beginning of the game. The deck is a random subset of a larger library of 120 cards. In particular, at the beginning of the game, the deck could have anywhere from zero to eight wizards , which are special, powerful cards. It is important for strategy to have a sense of how many wizards are in the deck. The game has 15 rounds, and during each round, some number of cards are revealed. The first round reveals $1\times 6$ random cards, the next reveals $2\times 6$ random cards, up to the final round which reveals all $15\times 6 = ...