What is the probability...?

In order to win this game, you must have an equal number of left and right moves. The barrier actually doesn't affect the outcome, because if you are in the middle, you just have to away from the middle anyway, even if there isn't a barrier. There are 14 levels. If we think of flipping a coin to determine direction, we want to find the probability of having exactly 7 of them be the same direction.

P(X = 7) = C(14,7) * ½^7 * ½^7

P(X = 7) = 3432 * ½^14

P(X = 7) = 3432 / 16384 = 429 / 2048

P(X = 7) ≈ 0.20947 ≈ 20.9%

For the second question, it depends on the number of levels. First, you need an even number of levels or else the penny will never be in the middle.

The generalized probability for 2k levels would be:

P(X = k) = C(2k, k) * ½^(2k)

The probability would get smaller and smaller as the number of levels increased.

Examples:

2 levels = 50%

4 levels = 37.5%

6 levels = 31.3%

8 levels = 27.3%

10 levels = 24.6%

12 levels = 22.6%

14 levels = 20.9%

16 levels = 19.6%

18 levels = 18.5%

20 levels = 17.6%

...

40 levels = 12.5%

...

60 levels = 10.3%

...

80 levels = 8.9%

...

100 levels = 8.0%

Another way to understand your answer is C(14,7) successful paths out of 2^14 possible paths C(14,7)/2^7.

However C(14,7) will count paths like LRRRRRRRLLLLLL (L=left move, R=right move) which cross the boundary, ie. It will over count.

The puzzle has more depth than on first appearance.

Hmm...I meant that to be a "comment".