project_euler.problem_121.sol1

A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted. After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random.

The player pays £1 to play and wins if they have taken more blue discs than red discs at the end of the game.

If the game is played for four turns, the probability of a player winning is exactly 11/120, and so the maximum prize fund the banker should allocate for winning in this game would be £10 before they would expect to incur a loss. Note that any payout will be a whole number of pounds and also includes the original £1 paid to play the game, so in the example given the player actually wins £9.

Find the maximum prize fund that should be allocated to a single game in which fifteen turns are played.

Solution:

For each 15-disc sequence of red and blue for which there are more red than blue, we calculate the probability of that sequence and add it to the total probability of the player winning. The inverse of this probability gives an upper bound for the prize if the banker wants to avoid an expected loss.

Functions

solution(→ int)

Find the maximum prize fund that should be allocated to a single game in which

Module Contents

project_euler.problem_121.sol1.solution(num_turns: int = 15) → int

Find the maximum prize fund that should be allocated to a single game in which fifteen turns are played. >>> solution(4) 10 >>> solution(10) 225