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What is the binomial distribution (and why it runs Plinko)?

Article updated: Portrait of KirillKirill

For a fair 50/50 bounce the formula is:

P(k) = C(n, k) / 2ⁿ
2ⁿ is the number of possible paths through n rows; C(n, k), "n choose k", counts the paths with exactly k rights.

Example - 16 rows. There are 2¹⁶ = 65,536 possible paths. Only one path goes right every time, so the far-right slot has probability 1/65,536. The center slot (8 rights, 8 lefts) can be reached by C(16, 8) = 12,870 paths - probability 19.64%. Together, the middle five slots (6 to 10) take 79.0% of all drops.

This is why the payout tables look the way they do. Slots the ball rarely reaches carry big multipliers; slots it reaches most of the time pay little. The probabilities never change between casinos or risk levels - only the payouts do.

In ProvablyHero tools

Every number on the odds pages and in the calculator comes from this formula.

  • What is an edge slot in Plinko?Edge slots are Plinko's leftmost and rightmost slots: the rarest landing spots and usually the biggest multipliers. Exact odds for 8, 12 and 16 rows.
  • What is a multiplier?A multiplier is what an outcome pays relative to your bet. Why most Plinko multipliers are below 1×, and how multipliers and probabilities make RTP.
  • What is RTP (return to player)?RTP is the share of all bets a game pays back over the long run. How to calculate it from a Plinko payout table, and what RTP can't tell you.
  • What is variance?Variance measures how widely results spread around the average. Why high-risk Plinko swings harder, how it relates to RTP, and how to see it in a simulator.