Question

If a fair coin flip comes up heads, you win a dollar, otherwise, you lose a dollar. You start with N dollars, and play repeatedly until you either reach M dollars or lose all your money, whichever comes first. M and N are fixed positive integers such that 0 < N < M.

(a) Find the expected length of the game if the coin is fair.

(b) Find the expected length of the game if the coin is biased, in other words, if the probability of heads is p ≠ 1/2.

Answer 1

To find the expected length of the game, we set up an equation and solve it recursively. If the coin is biased, the equation can be modified accordingly.

Step-by-step explanation:

To find the expected length of the game, we can use probability. Let's denote the expected length as E(N), where N is the initial amount of money. In each round, the probability of winning is 0.5 and the probability of losing is also 0.5. If you win, you gain $1, so your new amount of money becomes N + 1. If you lose, you lose $1, so your new amount of money becomes N - 1. We can set up the following equation:

E(N) = 0.5*(E(N+1) + 1) + 0.5*(E(N-1) + 1)

Simplifying this equation, we get: E(N) = 0.5*E(N+1) + 0.5*E(N-1) + 1

Now we can solve this equation recursively to find the expected length of the game for different values of N.

For part (b) of the question, if the coin is biased with probability of heads as p, we can modify the equation as follows: E(N) = p*(E(N+1) + 1) + (1-p)*(E(N-1) + 1). We can solve this equation recursively to find the expected length of the game for different values of p.