Answer:
219/256 = 0.855
Step-by-step 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.
