How Keno Works

Game Overview · N, D, s?

Start with the game configuration, choose an allowed set of numbers, wait for the draw, count the matches, and then read the probability for that match count.

Probability · P(K=k)?

P(K=k | N,D,s) = C(D,k) × C(N−D,s−k) / C(N,s)

Probability · P(K=5 | N=56,D=16,s=5)?

56/16 · s=5 · k=5 · P(K=5)=0.11435106% · ≈ 1:874.5

Game Overview · N, D, s

Editorial inference Start with the game configuration, choose an allowed set of numbers, wait for the draw, count the matches, and then read the probability for that match count.

Rules

Editorial inference Game configurations and prize tables are specific to each operator and offering, so consult the official rules for the exact game.

Number Pool → Picks Allowed → Numbers Drawn

Editorial inference Hypothetical example: a game draws 20 numbers from 80. You choose 2, 17, 31, 48, 76. The drawn numbers include 17, 48; the other 18 drawn numbers are not in your selection. That is exactly 2 matches from 5 picks.

Editorial inference Assuming equally likely draws without replacement: Exactly 2 matches when you pick 5 numbers: about 1 in 3.7. The probability is 27.045739%.

Editorial inference This match count does not guarantee a prize. Any payout depends on the game's official prize table and rules; this example does not validate a real ticket.

Editorial inference Number Pool=N · Picks Allowed=s · Numbers Drawn=D · Matches=k

Editorial inference N → S, |S|=s → R, |R|=D → K=|S∩R| → P(K=k)

NIST · H(N,D,s,k)

Editorial inference Exact-match probabilities follow the hypergeometric model. [1]

Probability · P(K=k)

Editorial inference A match is a number you selected that also appears among the drawn numbers, so K can run from zero through s.

Editorial inference Number Pool=N · Numbers Drawn=D · Picks Allowed=s · Matches=k

Editorial inference P(K=k | N,D,s) = C(D,k) × C(N−D,s−k) / C(N,s)

Probability Distribution · 56/16 · s=5

Editorial inference 56/16 · s=5 · k=0 · P(K=0)=17.22617006% · ≈ 1:5.81

Editorial inference 56/16 · s=5 · k=1 · P(K=1)=38.2803779% · ≈ 1:2.61

Editorial inference 56/16 · s=5 · k=2 · P(K=2)=31.03814425% · ≈ 1:3.22

Editorial inference 56/16 · s=5 · k=3 · P(K=3)=11.43510577% · ≈ 1:8.75

Editorial inference 56/16 · s=5 · k=4 · P(K=4)=1.90585096% · ≈ 1:52.47

Editorial inference 56/16 · s=5 · k=5 · P(K=5)=0.11435106% · ≈ 1:874.5

Editorial inference In the 56/16 example with five picks, the six rows K=0 through K=5 each show the chance of exactly that many matches and together cover every outcome.

Game Config · P(K=5 | s=5)

Editorial inference 56/16 · s=5 · k=5 · P(K=5)=0.11435106% · ≈ 1:874.5 · 70/20 · s=5 · k=5 · P(K=5)=0.12810032% · ≈ 1:780.64 · 80/20 · s=5 · k=5 · P(K=5)=0.06449247% · ≈ 1:1550.57

Probability ≠ Top prize

Editorial inference Probability alone does not state a payout; assessing financial value also requires the stake and the official paytable.

Editorial inference Probability=P(K=k) · Top prize=v(s)

Editorial inference P(K=k) ∈ [0,1] · V = Σ v(k) × P(K=k) · v(k)=∅ ⇒ V=∅

Keno Directory · Blog · Tools

• Keno Strategies: What the Mathematics SupportsNo number-selection system changes the probabilities fixed by the pool, draw count and picks. Exact-match probabilities follow the hypergeometric model.
• History of KenoAcademic research documents gambling and lottery practices in late-Qing China; it does not verify the Keno origin legend. The Nevada Historical Society documents Chinese lottery activity in Reno and the use of the name Race Horse Keno in the 1930s and 1940s.
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