Problem 10 Solution: "All Decked Out"

Let G be the average number of coins on the cards in your deck (both treasure cards and +coins action cards) and let C be the average number of +cards on cards in your deck (i.e. 0 for treasure cards and action cards with only +coins, or X for action cards with +X cards).

Let P be the expected value of a single card. On average, a single card gives you G gold plus C new cards, and the expected value of those C new cards is P each. So we have P = G+CP, or (1-C)P = G, or P = G/(1-C). This means that the expected value of a five card draw is 5G/(1-C).

Note that if C is equal to or greater than 1, this means that on average you will be able to keep drawing cards indefinitely, and our assumption that you will never run out of cards in your deck would be inaccurate.