B. Hughes, 2017 August 27

The above graph represents the probability estimates obtained by playing 1,280 games of Klondike solitaire using cards for 260 of the games and a computer for other 1,020. Two curves have been fitted to the result:

Probability (n) = 0.0236n + 0.0404…………… 0 ≤ n ≤ 3 cards
Probability (n) = 0.1025 e^-0.1808(n-4)………..4 ≤ n ≤ 40 cards

with

Probability (n) = 0……………………………….… 41 ≤ n ≤ 51 cards
Probability (n) = 0.0781………………………………….n = 52 cards

1. Introduction

It has been determined that the theoretical probability of winning an open game of Klondike solitaire, i.e. all 52 cards in the foundation stacks at the end of the game, and all cards known to the player at all times in the game, is approximately¹ 0.8. Anyone who has played a traditional non-open game – cards face down in the tableau except for the end card in each column, and the remaining cards hidden from view except on turnover –  knows that the probability of winning is more like a tenth of this². The number of different card arrangements is 52! (approx. 10⁶⁸) and for the traditional Klondike games there are many more possibilities than this because of choices that arise during each game. The problem of determining the probability of winning in a traditional Klondike game is very difficult and has not yet been done. Nevertheless, by playing a large number of games, estimates of the statistics can be determined.   The present report provides an estimate of the complete probability density function (pdf) for the number of cards in the foundation stacks at the end of a game, i.e. the probability of ending with zero cards, or one card, or two cards, etc. or a “win” of 52 cards. It is recognized that some players are more adept at the game than others but ……(more)