Expected Winnings When Playing Keno
Overview
It's common knowledge that the entity that runs a gambling game has set it up so they don't lose in the long term. Not all states' Keno games are created equal, though. Below is a detail of long-term expected winnings (technically, all are losses) when playing Keno in different states along with the calculation methodology.
Summary
Table 1. Expected long-term earnings per draw in various states. The game highlighted green is the best choice (least loss). The 1-Spot game is typically the worst, so the spot highlighted red is the 2nd-worst.
| State | 10-Spot | 9-Spot | 8-Spot | 7-Spot | 6-Spot | 5-Spot | 4-Spot | 3-Spot | 2-Spot | 1-Spot | Winnings Page |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Kentucky | -$0.36 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.34 | -$0.50 | Link |
| Maryland | -$0.43 | -$0.44 | -$0.41 | -$0.40 | -$0.44 | -$0.46 | -$0.42 | -$0.38 | -$0.40 | -$0.50 | Link |
| Michigan | -$0.36 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.34 | -$0.50 | Link |
| Ohio | -$0.36 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.35 | -$0.34 | -$0.50 | Link |
| Virginia | -$0.30 | -$0.27 | -$0.27 | -$0.28 | -$0.28 | -$0.30 | -$0.26 | -$0.31 | -$0.28 | -$0.25 | Link |
Methodology
Tables 2, 3, and 4 below are based on Michigan's payout breakdown.
- Find the state's published winnings for each combination of spots. This is the only variable since all win probabilities are exactly the same if the game is truly random.
- Transpose the winnings into a table. This isn't required; it just makes it easier to visualize.
Table 2.Winnings breakdown for matching R numbers in N-Spot game.N-Spot Game Matched Numbers (R) 10 9 8 7 6 5 4 3 2 1 0 10 100,000 5,000 500 50 10 2 0 0 0 0 5 9 - 25,000 2,000 100 20 5 2 0 0 0 0 8 - - 10,000 300 50 15 2 0 0 0 0 7 - - - 2,000 100 11 5 1 0 0 0 6 - - - - 1,100 57 7 1 0 0 0 5 - - - - - 410 18 2 0 0 0 4 - - - - - - 72 5 1 0 0 3 - - - - - - - 27 2 0 0 2 - - - - - - - - 11 0 0 1 - - - - - - - - - 2 0
- Calculate the probability of matching R numbers of the N-Spot game. The probability = [number of possible ways that the R matched numbers can be chosen from the 20 in the draw] x [number of possible ways that the unmatched numbers can be chosen from the 60 that weren't drawn] / [total number of combinations of N numbers among the 80 possible]. P = 20CR · 60CN-R / 80CN.
Table 3. Probability of choosing R correct numbers in N-Spot game.N-Spot Game Matched Numbers (R) Win Ratio 10 9 8 7 6 5 4 3 2 1 0 10 0.00% 0.00% 0.01% 0.16% 1.15% 5.14% 14.73% 26.74% 29.53% 17.96% 4.58% 1:9.05 9 - 0.00% 0.00% 0.06% 0.57% 3.26% 11.41% 24.61% 31.64% 22.07% 6.37% 1:6.53 8 - - 0.00% 0.02% 0.24% 1.83% 8.15% 21.48% 32.81% 26.65% 8.83% 1:9.77 7 - - - 0.00% 0.07% 0.86% 5.22% 17.50% 32.67% 31.52% 12.16% 1:4.23 6 - - - - 0.01% 0.31% 2.85% 12.98% 30.83% 36.35% 16.66% 1:6.19 5 - - - - - 0.06% 1.21% 8.39% 27.05% 40.57% 22.72% 1:10.34 4 - - - - - - 0.31% 4.32% 21.26% 43.27% 30.83% 1:3.86 3 - - - - - - - 1.39% 13.88% 43.09% 41.65% 1:6.55 2 - - - - - - - - 6.01% 37.97% 56.01% 1:16.63 1 - - - - - - - - - 25.00% 75.00% 1:4.00
- The expected value is just the probability of that match-spot combination multiplied by the winnings earned at the match-spot. The expected winnings is the summation of expected values subtracted from the original bet ($1).
Table 4. Expected winnings per $1 draw for each N-Spot game.N-Spot Game Matched Numbers (R) Expected
Winnings10 9 8 7 6 5 4 3 2 1 0 10 0.01 0.03 0.07 0.08 0.11 0.10 - - - - 0.23 -$0.36 9 - 0.02 0.07 0.06 0.11 0.16 0.23 - - - - -$0.35 8 - - 0.04 0.05 0.12 0.27 0.16 - - - - -$0.35 7 - - - 0.05 0.07 0.10 0.26 0.17 - - - -$0.35 6 - - - - 0.14 0.18 0.20 0.13 - - - -$0.35 5 - - - - - 0.26 0.22 0.17 - - - -$0.35 4 - - - - - - 0.22 0.22 0.21 - - -$0.35 3 - - - - - - - 0.37 0.28 - - -$0.35 2 - - - - - - - - 0.66 - - -$0.34 1 - - - - - - - - - 0.50 - -$0.50
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