The Problem of the Points. Pascal and Fermat also explored a question concerning how to divide the stakes in a game that has been interrupted. Suppose two players, Blaise and Pierre, are playing a game where the winner is the first to achieve a certain number of points. The game gets interrupted at a moment when Blaise needs n more points to win and Pierre needs m more to win. How should the game’s prize money be divvied up? Fermat argued that Blaise should receive a proportion of the total stake equal to the chance he would have won if the game hadn’t been interrupted (and Pierre receives the remainder).
Assume the game is played in rounds, the winner of each round gets 1 point, rounds are independent, and the two players are equally likely to win any particular round.

a. Write a program to simulate the rounds of the game that would have happened after play was interrupted. A single simulation run should terminate as soon as Blaise has n wins or Pierre has m wins (equivalently, Blaise has m losses). Use your program to estimate P(Blaise gets 10 wins before 15 losses), which is the proportion of the total stake Blaise should receive if n = 10 and m = 15. 

b. Use your same program to estimate the relevant probability when n = m = 10. Logically, what should the answer be? Is your estimated probability close to that?
c. Finally, let’s assume Pierre is actually the better player: P(Blaise wins a round) = .4. Again with n = 10 and m = 15, what proportion of the stake should be awarded to Blaise?