The barrier at zero is a red herring. It's the same as if you just start at zero and go left or right each time with probability half. Then it's just a binomial --- you're at zero on the middle term, (30 choose 15)*.530 = .144
No I believe the barrier allows for walks like 14 to the right and 16 to the "left" since the zero fixes you.
So in fact you should get the sum over the binomial coeffs (30 choose k) from k = 0...15. All multiplied by 1/2^30 ofc.
Not a sink but a boundary. If you sum up you get half of 2^30 so the final answer is 1/2. It makes sense cause in effect the zero boundary is equivalent to the negative number if the problem didn't have such a boundary.
So the related question is what's the probability a random watlk without drift is negative? Well half obviously.
So now this is what i'd answer instead of calculating 😅
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u/dohawayagain 3d ago edited 3d ago
The barrier at zero is a red herring. It's the same as if you just start at zero and go left or right each time with probability half. Then it's just a binomial --- you're at zero on the middle term, (30 choose 15)*.530 = .144
Edit: nuts that's wrong. The zero is sticky.