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## Question

Alternate approach to the ant-checkerboard problem asks: What is the probability of flipping 8 heads before 5 tails?

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Spoiler

Flipping a coin eleven times results in 11C8+11C9+11C10+11C11 or 232 trials that have 8 or more heads.  There are 11C7 or 330 trials that would have exactly seven heads, half of which would have eight heads with one more flip:

(11C7/2+11C8+11C9+11C10+11C11)/211=.193848

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Naive again

12C8/4096 = .12084961

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8 hours ago, CaptainEd said:

Naive again

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12C8/4096 = .12084961

That was the first-cut answer in the ant-chessboard problem.

Spoiler

The value is too low. It counts only once the (multiple) cases where 8 heads is reached before the "full complement" of 4 tails occurs.

The solution is easier to find if we (reasonably) assume that

Spoiler

the final toss will be Heads.

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Oops! Good point!

Ok, 11C7/2048 = 0.1611328125

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17 hours ago, plainglazed said:
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Flipping a coin eleven times results in 11C8+11C9+11C10+11C11 or 232 trials that have 8 or more heads.  There are 11C7 or 330 trials that would have exactly seven heads, half of which would have eight heads with one more flip:

(11C7/2+11C8+11C9+11C10+11C11)/211=.193848

Spoiler

Equates with the 3x speed version of ants on a checkerboard:
probability of reaching either of the ends of the five meeting points = 794 / 4096

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