Question

Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with $256, find the probability that after 8 bets, he will have at least $50.

Answer 1

219/256 = 0.855

Explanation:

He starts with $256.

After the first bet, he either has $128 or $512.

After the second bet, he either has $64, $256, or $1024.

Repeating this pattern, the amount after n bets and r wins is:

A = 256 (½)ⁿ⁻ʳ (2)ʳ

And the number of ways A can be achieved is:

N = nCr

We're given that n = 8.

A = 256 (½)⁸⁻ʳ (2)ʳ

If r = 0, A = 1 and N = 1.

If r = 1, A = 4 and N = 8.

If r = 2, A = 16 and N = 28.

If r > 2, A > 50. The sum of N for all r is 2⁸ = 256.

Of the 256 possible combinations, 1+8+28 = 37 result in less than $50, so 219 result in more than $50.

So the probability is 219/256 = 0.855.