Plinko odds

One bounce, one coin flip

At every peg the ball goes left or right with equal chance. After r rows it has taken k right bounces, and the slot it lands in is just k. The number of paths that produce k rights out of r bounces is the binomial coefficient, so:

P(slot k) = C(r, k) / 2^r

On 16 rows that gives 17 slots with these probabilities from the centre outward:

Nearly one drop in five lands dead centre; the two edge slots together catch three drops in a hundred thousand. Everything about Plinko's feel follows from that curve.

What the risk level changes

Rows (8, 10, 12, 14 or 16) set the shape of the curve. Risk (Low, Medium, High) sets the paytable laid over it. On 16 rows the edge slot pays 16× on Low, 110× on Medium and 1,000× on High, while the centre slot pays 0.5×, 0.3× and 0.2×. High moves value out of the common middle and into the rare edges; the total expected return is the same on every setting because each table is solved to the same house edge.

expected return = Σ_k P(slot k) × payout(k)   — held constant across Low / Medium / High

Why fewer rows feel calmer

Eight rows means nine slots and a much flatter curve: the edge is 1 in 256 instead of 1 in 65,536, so the top multiplier is correspondingly smaller (29× on High). If you want frequent small wins, fewer rows and Low risk; if you are hunting one screenshot, 16 rows and High. Neither is cheaper.

Provably fair bounces

The path is not simulated physics. Each bounce is a bit taken from the HMAC-SHA256 of the server seed, your client seed and the bet nonce; for more than 8 rows a second hash (nonce suffixed with :1) supplies the extra bits. The animation replays a decision that already exists, which is why you can verify it afterwards.