A binomial distribution doesn't predict any single trial. It counts how many successes pile up across n independent attempts. Each ball below falls through 12 pegs — at every peg a fresh trial with the same chance p of bouncing right.
Four things make a setup binomial: a fixed number of trials n fixed up front; each trial has just two outcomes — success or failure; the success probability p is identical every trial; and the trials are independent, so one result can't sway another. Every peg below is a fresh trial with the same p, and no ball remembers the ones before it — which is why the same lawful shape builds every time.
P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ. The C(n,k) is pure counting — how many distinct paths land exactly k successes. More routes exist to the middle than the edges, which is why the distribution always bulges at μ = np.