A small deck of cards has cards of three types, Reshuffle, One and Zero.
- Drawing a Reshuffle card scores 0. Then all drawn cards are replaced back into the deck, and the deck is shuffled.
- Drawing a One card scores 1 point. Drawn One cards are retained in hand until the next Reshuffle is drawn.
- Drawing a Zero card scores 0. Drawn Zero cards are retained in hand until the next Reshuffle is drawn.
- A. The deck contains 1 One and 2 Reshuffle cards. Draw 100 times. What is the expected score? (25, scoring 1 per 4 draws from 3 cards)
- B. The deck contains 1 One, 10 Zero and 2 Reshuffle cards. Draw 100 times. What is the expected score? (7.1, i.e scoring 1 per 14 draws from 13 cards)
- C. The deck contains i One, j Zero and k Reshuffle cards. What's the formula for the expected score? (No idea)
I've run a Monte-Carlo simulation of this, and am reasonably sure of the answers to A and B. My maths isn't up to rigorously defining C.
This is a representation of the attack deck in the board game Gloomhaven. The deck always contains two cards which cause a reshuffle, plus some variable number of other cards. The Monte-Carlo simulation was intended to work out how often a particular single special card would be drawn, here represented by the One. I found the conclusion was interesting and surprising. I'm asking here to confirm that result.
I can see that the answer to A (1 in 4 draws are the One, total average score for 100 thus being 25) comes from there being four different draw possibilities - the obvious initial three, and the fourth when the One is already in hand and the next draw must be an unscoring reshuffle. But I don't know how to represent or derive that, and I would like an answer to C that I can grasp as a none-mathematician.