How Is Keno Probability Calculated? 40 Numbers, 10 Draws, and the Hypergeometric Distribution

Direct answer

Keno is a classic sampling-without-replacement problem. Pick count, draw count and exact hits fit directly into the hypergeometric distribution.

Quick facts

Population

40 numbers

Drawn

10 numbers

Player picks

1–10

Exactly h hits

Hypergeometric

01

The hypergeometric formula

A player chooses n numbers, then 10 numbers are drawn without replacement from 40. The probability of exactly h hits is P(H=h)=C(n,h)C(40−n,10−h)/C(40,10).

The first combination chooses the h winning numbers from your n picks; the second chooses the remaining 10−h drawn numbers from the 40−n numbers you did not pick.

02

Why this is not exactly binomial

Every draw reduces the remaining pool, so the trials are not independent draws with replacement. That is the key distinction between hypergeometric and binomial models.

Treating every draw as an independent fixed n/40 hit chance gives an approximation rather than the exact distribution.

03

Expected number of hits

The full distribution is hypergeometric, but the expected hit count is simple: E[H]=n×10/40=n/4.

If you pick 8 numbers, the average is 2 hits. A single round can still produce 0, 1, 2 or more.

04

Hit probability is not the same as payout

The combinatorics determine the hit distribution. RTP also depends on the paytable multiplier assigned to every possible hit count h.

Evaluating a Keno table therefore requires the weighted sum of all P(H=h) × multiplier(h), not just the top jackpot or highest-hit multiplier.

EXAMPLE

Picking 5 numbers

• E[H] = 5×10/40 = 1.25 hits
• Exactly h hits: C(5,h)C(35,10−h)/C(40,10)
• Theoretical RTP: Σ P(H=h) × payoutMultiplier(h)

Frequently asked questions