# Keno Probability: How to Increase Keno Winnings

TR Smith is a product designer and former teacher who uses math in her work every day.

Keno is a relatively simple lottery game that is offered as an instant lottery game in many places where they sell lotto tickets, as well as in some casinos. Since it is a game of pure chance, there is no special trick to picking matching numbers. But there is an advantage in knowing which version of the game to play, i.e., the number of spots, as some spot versions yield higher average payouts and higher odds of winning than others. The degree to which some spot versions are more lucrative than others depends on the payout structure, which varies by state or province.

Popular logos advertising keno games.

What are the versions? In every Keno game, a computer randomly chooses 20 distinct numbers from 1 to 80. The keno player bets money on whether he can match 1 of those 20 numbers, 2 out of 20, etc. all the way up to 10 out of 20. If you place a bet that you can match 5 out of 20 numbers, you are playing the "5-spot" version. The big question for the keno player is which spot version gives the highest probability of winning any prize, and which spot version yields the highest return per \$1 wagered? The answers to these two questions depend on how each state's lottery commission sets keno prizes, but regardless of where you play keno, the answer is found by calculating the keno probabilities.

Instant lottery games like keno are popular among slot machine players.

## 5-Spot Keno Probabilities

The 5-spot version is one of the most popular among keno players. When you play the 5-spot game you pick 5 numbers from 1 to 80, and then see if some or all of them match numbers in a set of 20 numbers that the computer picks from 1 to 80. In Ohio, for every \$1 wager the payouts are awarded according to the following prize table:

5 out of 5 match: win \$410
4 out of 5 match: win \$18
3 out of 5 match: win \$2

First, let's calculate the probability of winning any prize. To do this, we compute the probabilities of matching exactly 5 out of 5, exactly 4 out of 5, and exactly 3 out of 5, and then add them.

P(5 out of 5) = (5 C 5)*(75 C 15)/(80 C 20)
= 51/79079
= 0.00064492

P(4 out of 5) = (5 C 4)*(75 C 16)/(80 C 20)
= 3825/316316
= 0.012092

P(3 out of 5) = (5 C 3)*(75 C 17)/(80 C 20)
= 13275/158158
= 0.083935

In the expressions above, the function (x C y) is the combination function "x choose y" or the number of ways to choose y objects out a a set of x objects. The total probability of winning any prize is the sum of these three probabilities, 30579/316316 or 0.096672. As a percent, the chances are 9.6672%. As odds, the chances are 1 in 10.3442.

In general, if you are playing the n-spot keno game, the probability of matching k out of n numbers is given by the formula

P(k out of n) = (n C k)*(80-n C 20-k)/(80 C 20)

## 5-Spot Keno Expected Winnings per \$1 Wager

Knowing that the overall chance of winning a prize in 5-spot keno game is 1 out of 10.3442 gives you an idea of how frequently you will win something. But what gamblers really want to know is how much they can expect to win for every \$1 they play. To calculate the expected payout for the 5-spot keno game in Ohio, you simply multiply each probability by the payout and add up the numbers.

P(5 out of 5)*\$410 + P(4 out of 5)*\$18 + P(3 out of 5)*\$2
= 0.00064492*\$410 + 0.012092*\$18 +0.083935*\$2
= \$0.6499432

On average, the 5-spot keno game in Ohio pays about 65 cents on the dollar.

Many gamblers don't understand the math behind games of chance like keno. Smart gamblers know how to compute their odds of winning and expected earnings.

## Which Spot Game is Best?

To figure out which spot version of the keno game will maximize your expected earnings, you need to compute the expected winnings for the 1-spot, 2-spot, 3-spot, ... 10-spot games following the mathematical routine explained above. Because prizes vary by state, the results you come up with for your state may not be the same for a keno player living elsewhere. Every state has a lottery commission that publishes payout structures on their website.

For example, in Ohio, the 1-spot game has a 1 in 4 chance of winning, but only pays 50 cents on the dollar. That means compared to the 5-spot game, you will win more often playing the 1-spot game, but in the long run you will lose more money. The 4-spot game gives a 1 in 3.68 chance of winning, and the expected earnings per \$1 wager are \$0.649439. This is just slightly less than the expected winnings of the 5-spot game. By computing the expected winnings for ever spot version in Ohio, you will see that the 5-spot game does yield the highest expected winnings, so if you live in Ohio and like to play keno, that is the version you should play.

## Playing Responsibly

If you like lotto games, it's crucial that you understand how to compute your chances of winning a prize and what your expected winnings per \$1 wagered will be. All lottery games and casino games of chance are structures so that the "house wins," meaning that in the long run, your expected earnings per \$1 wagered will always be less than \$1.

By knowing how to compute gambling probabilities you can manage your expectations and avoid betting more than you can afford to lose.

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