Why a spreadsheet is the right tool for keno odds
Keno looks simple: pick some numbers, watch the draw, and see how many match. The odds underneath that simplicity are combinatorial, and they change sharply with every number you choose. A spreadsheet is a natural fit because the same formula, copied across a grid, can show you the probability of matching zero, one, two, or every spot you picked. Before you open Excel, it helps to know what the formula is counting.
Keno is a lottery-style draw. The operator draws a fixed set of numbers from a larger pool, and you have marked your own set on a ticket. The question the odds answer is: given how many numbers you picked and how many the draw contains, what is the probability of matching exactly some number of them?
That probability comes from a hypergeometric distribution. The draw selects numbers without replacement, which means each number can appear only once. Excel has a built-in function for exactly this situation, and once you understand its four arguments, the rest of the sheet is just layout.
The combinatorics behind a keno match
Think of the draw as splitting the number pool into two groups: the numbers you picked and the numbers you did not. To match exactly k of your spots, the draw must contain k numbers from your group and the remainder from the group you left unpicked.
Three quantities define the whole calculation. The pool size is how many numbers the draw selects from. The draw size is how many numbers are drawn. The spot count is how many numbers you marked. From those three, every probability in keno follows.
The number of ways to choose k matching numbers from your spots is a combination. The number of ways the draw fills its remaining positions from the numbers you did not pick is another combination. Multiply those, then divide by the total number of possible draws from the full pool. That quotient is the probability of exactly k matches.
For a concrete example, suppose the pool has 80 numbers, the draw takes 20, and you picked 4 spots. The probability of matching exactly 2 of them is the number of ways to choose 2 from your 4, multiplied by the number of ways to choose the other 18 drawn numbers from the 76 you did not pick, all divided by the number of ways to choose 20 from 80.
Building the formula in Excel
Excel's HYPGEOM.DIST function performs this calculation directly. Its arguments are, in order: the number of matches you are testing, the draw size, the number of spots you picked, the pool size, and a cumulative flag. Set the last argument to FALSE for the probability of exactly that many matches, or TRUE for the probability of that many or fewer.
Set up four input cells at the top of your sheet: pool size, draw size, spot count, and the match count you want to test. Then the formula reads something like =HYPGEOM.DIST(match_cell, draw_cell, spot_cell, pool_cell, FALSE). Copy it down a column of match counts from zero up to your spot count, and you have a complete probability table for that ticket.
One caution: HYPGEOM.DIST returns an error if the match count exceeds the spot count or the draw size. Your table should therefore stop at the smaller of those two numbers. If you picked 10 spots but the draw only takes 20 from a pool of 80, matching all 10 is possible and the formula handles it, but matching 11 is not a valid input.
From probabilities to odds and expected value
A probability is not the same as the odds a keno pay table quotes. Probability is the share of all possible draws that produce a given match count. Odds in the everyday sense are usually expressed as a ratio of unfavourable to favourable outcomes. To convert a probability p to that ratio, compute (1-p)/p and read it as that many to one.
Expected value goes one step further. It asks what a ticket returns on average per unit wagered, given the pay table. For each match count, multiply the probability by the payout for that count, then sum across all possible counts. If the result is below 1, the ticket returns less than it costs on average, which is the normal structure of a keno pay table.
You can build this in the same sheet. Put the payouts in a column next to the probabilities, add a column for the product of the two, and sum that column. The result is the expected return per unit staked, under the specific pay table you entered. Change the pay table and the expected value changes, even though the probabilities stay fixed.
What changes when you change the spot count
The spot count is the one variable you control before the draw, and it reshapes the whole probability table. More spots do not simply raise or lower every match probability in the same direction. They shift the distribution.
The table below compares a low spot count with a high one, using a pool of 80 and a draw of 20 as an example.
| Property | Low spot count (e.g. 3) | High spot count (e.g. 10) |
|---|---|---|
| Most likely match count | 0 or 1 | 2 or 3 |
| Probability of matching all spots | Higher per spot | Far lower |
| Probability of matching none | Lower | Higher |
| Spread of outcomes | Narrow | Wide |
| Typical pay table emphasis | Smaller, steadier payouts | Large payout for many matches |
The exact numbers in each cell come from the formula you have already built. The structural point is that the spot count changes which match counts are probable, and the pay table decides whether those probable outcomes return enough to make the ticket worthwhile as a long-run proposition. No spot count makes a negative-expectation pay table positive; it only changes the shape of the losses.
Checking your sheet against a known case
A spreadsheet that produces numbers is not automatically right. The fastest check is to sum the probabilities for every possible match count. They must total exactly 1, because one of those outcomes always occurs. If the sum is off, check the pool size, draw size, and spot count cells for a mismatch.
A second check is the cumulative flag. The probability of matching at most k numbers, computed with TRUE, should equal the sum of the exact probabilities from zero through k. If those two columns disagree, the formula is pointing at the wrong cells.
For a third check, change the spot count to 1 and the match count to 1. The probability should reduce to the draw size divided by the pool size, because with one spot you simply need that number to appear in the draw. If your formula does not return that ratio, one of the arguments is in the wrong position.
FAQ
Can I use the same formula for keno games with different pool sizes?
Yes. The pool size and draw size are input cells, not constants. Change them to match the game you are analysing, and every probability in the table recalculates. The combinatorics are identical; only the numbers fed into the formula differ.
Does the order of the drawn numbers matter in keno odds?
No. Keno matches are counted as sets, not sequences. The hypergeometric distribution treats all selections without regard to order, which is why a single combination formula covers every match count rather than a more complex permutation calculation.
Why does my expected value stay below 1 no matter which spots I pick?
Because the pay table is built that way. The probabilities are fixed by the game's structure, and the payouts are set so that the sum of probability times payout, across all match counts, is below the cost of the ticket. Changing spots redistributes the probabilities but does not change the fact that the pay table returns less than the stake on average.
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