# Keno Lottery Strategies, Odds, & Expected Winnings

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

Keno is a number drawing lottery game offered by many states. Keno games are fast since computerized drawings are held every 4 minutes or so and displayed on a monitor at the keno lottery vendor; players know within a few minutes if their numbers have won or lost.

How Keno Works: A random number generator outputs 20 distinct numbers drawn from 1 to 80. Before the "drawing" occurs, a player decides which n-spot game he would like to play. Each n-spot game can be played for a \$1 wager. In the 1-spot game a player chooses 1 number from 1 to 80; in the 2-spot game he chooses two different numbers from 1 to 80; in the 3-spot game he chooses three distinct numbers from 1 to 80; etc. In most states, n ranges from 1 to 10, that is, you can play the 1-spot, 2-spot, ..., or 10-spot game. In Massachusetts, n goes up to 12. You win money if you some of the numbers you select match some of the 20 numbers displayed on the keno screen. The closer you are to a perfect match of all n numbers, the more money you win. Partial matches pay less. In some n-spot games, you even win a prize for matching 0 out of n numbers, since statistically it is a rare event.

The natural question to ask is, what value of n offers the optimal winning strategy? In other words, which n-spot game maximizes the expected return on a keno lottery wager? The answer depends on the payouts for various matches, which varies by state. The method of computing the expected winnings is the same, regardless of the different payouts, so for this tutorial we will use the Massachusetts Keno Lottery as an example of how to determine the best keno lottery strategy.

## Massachusetts Keno Payout Table

The table below shows the prize amounts for matching some of the 20 numbers in a Massachusetts keno drawing. Notice that the 12-, 11-, and 10-spot games also pay if you manage to match 0 numbers. (Use scroll bar below table to see all Match/Win columns.)

Spots
Match/Win
Match/Win
Match/Win
Match/Win
Match/Win
Match/Win
Match/Win
Match/Win
12
12 numbers, win \$1,000,000
11 numbers, win \$25,000
10 numbers, win \$2,500
9 numbers, win \$1000
8 numbers, win \$150
7 numbers, win \$25
6 numbers, win \$5
0 numbers, win \$4
11
11 numbers, win \$500,000
10 numbers, win \$15,000
9 numbers, win \$1,500
8 numbers, win \$250
7 numbers, win \$50
6 numbers, win \$10
5 numbers, win \$1
0 numbers, win \$2
10
10 numbers, win \$100,000
9 numbers, win \$10,000
8 numbers, win \$500
7 numbers, win \$80
6 numbers, win \$20
5 numbers, win \$2
0 numbers, win \$2

9
9 numbers, win \$40,000
8 numbers, win \$4,000
7 numbers, win \$200
6 numbers, win \$25
5 numbers win \$5
4 numbers, win \$1

8
8 numbers, win \$15,000
7 numbers, win \$1,000
6 numbers, win \$50
5 numbers, win \$10
4 numbers, win \$2

7
7 numbers, win \$5,000
6 numbers, win \$100
5 numbers, win \$20
4 numbers, win \$3
3 numbers, win \$1

6
6 numbers, win \$1,600
5 numbers, win \$50
4 numbers, win \$7
3 numbers, win \$1

5
5 numbers, win \$450
4 numbers, win \$20
3 numbers win \$2

4
4 numbers, win \$100
3 numbers, win \$4
2 numbers, win \$2

3
3 numbers, win \$25
2 numbers, win \$2.50

2
2 numbers, win \$5
1 number, win \$1

1
1 number, win \$2.50

## Calculating Keno Probability and Odds

For an n-spot game, the probability that k of your n numbers match some of the 20 displayed on the keno screen is given by the function

Prob(n, k) = [(n C k)*(80-n C 20-k)]/(80 C 20)

where (x C y) is the combinatorial function "x choose y." For example, if you play the 5-spot game, the probability that 3 of your 5 numbers match is

Prob(5, 3) = [(5 C 3)*(75 C 17)]/(80 C 20)
= 10 * 29673694525643100 / 3535316142212174320
= 0.08393505
= 8.393505%
= odds of 1 in 11.914

Keno Machine: A device that turns cash into disappointment.

## Calculating Overall Chance of Winning (or not losing)

To figure the overall chance of winning a prize for an n-spot game, including any break-even \$1 prizes, you simply add up all the probabilities of winning for the all the k-out-of-n matches. When break-even prizes are included in the tally, you are technically calculating the odds of not losing, to be more precise. Let's use the 5-spot game as an example again.

We already saw in the previous section that the probability of matching 3 out of 5 in the 5-spot game is 0.08393505. The probability of matching 4 out of 5 is

[(5 C 4)(75 C 16)]/(80 C 20)
= 0.01209234

and the probability of matching 5 out of 5 is

[(5 C 5)(75 C 15)]/(80 C 20)
= 0.0006449247

There are no prizes for matching 0, 1, or 2 out of 5 numbers. Therefore, the total probability of winning anything in the 5-spot game is

0.08393505 + 0.01209234 + 0.0006449247
= 0.0966723147
= odds of 1 in 10.344

You can do the same calculation for the other n-spot games for every value of n from 1 to 12. This produces the following table of overall odds for each n-spot game in the Massachusetts keno lottery.

(click column header to sort results)
Spots
Overall Odds as 1 in ...
12
15.73
11
7.63
10
9.05
9
6.53
8
9.77
7
4.23
6
6.19
5
10.34
4
3.86
3
6.55
2
2.27
1
4.00

The game with the best overall chance of not losing is the 2-spot game, followed by the 4-spot game. The game with the worst overall chance of not losing is the 12-spot game, followed by the 5-spot game. One might think then that the 2- and 4-spot games are the least risky, while the 12- and 5-spot games are the most risky. But overall odds are only part of the story. What you should really look at is the expected return or expected winnings for each \$1 wager on an n-spot game. The optimal keno strategy is to play the value of n that yields the highest expected return.

## Calculating Expected Winnings

The expected winnings dollar amount is found by multiplying each k-out-of-n probability by the payout for that match, then summing those numbers. Let's use the 5-spot game once again as an example.

The probabilities for matching 3, 4, and 5 out of 5 numbers are respectively 0.08393505, 0.01209234, and 0.0006449247. The payouts for these matches are respectively \$2, \$20, and \$450. This means the expected winnings are

0.08393505*2 + 0.01209234*20 + 0.0006449247*450
= 0.6999330150 dollars

or about 70 cents. In other words, for every \$1 you wager, you can expect to receive about \$0.70 back. In the long run, you can expect to lose \$0.30 for every \$1 you wager on Massachusetts' 5-spot keno game.

The table below summarizes the expected earnings for every n-spot game in Massachusetts keno lottery.

(click column header to sort results)
Spots
Expected Winnings per \$1
12
\$0.6978
11
\$0.6912
10
\$0.6931
9
\$0.6978
8
\$0.6900
7
\$0.6996
6
\$0.6907
5
\$0.6999
4
\$0.6920
3
\$0.6934
2
\$0.6804
1
\$0.6250

As you can see, in the Massachusetts keno lottery, the 5-spot game offers the highest return on your wager, while the 1-spot game offers the least. Therefore, to maximize your earnings, or minimize your losses, you should play the 5-spot game. The differences in the table may seem negligible, but in aggregate if you take all the people who play the 5-spot game and all the people who play the 1-spot game, those who play the 5-spot game get significantly better returns on their wagers.

These figures are only for Massachusetts keno lottery. In other states, the payouts for prizes are different, and therefore the expected winnings are different. You can apply the same statistical analysis to your state's keno lottery to discover the optimal n-spot game.

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• IDONO 3 years ago from Akron Ohio

I'm not going to make this easy on you. You're pretty good and I suspect you enjoy a challenge.

My question is, are those really true, absolute odds? Here is why I ask.

If I pick 5 numbers at the start of the game and the first number drawn is not one of mine, does that change the odds since there are only 79 numbers left in the pool? 2nd not mine. Now only 78 in the pool. And so on. I'm wondering if the end result of your formula comes up with the same result in this context. Good Luck.

• Author

TR Smith 3 years ago from Eastern Europe

Thanks for the good question. It turns out the the probability is the same regardless of whether the keno machine displays all 20 numbers at the same time, or creates suspense by displaying the numbers one by one.

In the scenario you describe, what happens is a set of intermediate conditional probabilities arises, but these intermediate probabilities have no effect on the game because what only matters is what the full set of twenty numbers is.

Say you play the 5-spot game and you pick the numbers "1," "2," "3," "4," and "5." Then the keno screen highlights the number "23." The probability that your five numbers will appear in the set of twenty, given that "23" has already appeared, is a conditional probability. Now the computer highlights "5." Now the probability that you make a perfect match, given that "23" and "5" have already appeared, is another (and different) conditional probability. The previous conditional probability no longer matters. It continues until the twentieth number is drawn. When the last number is drawn you either have some kind of match or you don't, so it's 0 or 1, probability-wise. But this analysis would only matter if you could choose your numbers AFTER some numbers were displayed, which you can't.

As the formula shows, the odds are simply determined by the possible number of ways your n numbers can occur in a set of twenty, divided by the total number of ways twenty numbers can be drawn from eighty, without any consideration of intermediate displays when numbers are drawn one by one. If you're curious, the denominator in this expression is 3,535,316,142,212,174,320!

• Rick 3 years ago

Is it wise to pay twice the amount of the ticket for the bonus? It is a multiplier that can be 0, 3, 4, 5 or 10.

Also, I don't play anything more than a 5-spot game because winnings over \$600 (per game, per ticket) are taxable. On the rare occasions when you hit all of your numbers, it's far better to win under \$600, and better still if you have multiple winning tickets that are all under \$600.

• Author

TR Smith 3 years ago from Eastern Europe

Hi Rick,

I'm not sure which state offers these multpliers, but assuming the multiplier is equally likely to be 0, 3, 4, 5 or 10, then the average multiplier value is (0+3+4+5+10)/5 = 4.4. But you have to divide it by 2 because you are paying twice the cost of a ticket. So effectively you get 2.2 times the average prize on average.

That seems kind of high to me, so I wonder if the multipliers aren't weighted so that 0 occurs more often. Which state is this? They must publish the relative frequencies of the multipliers on their lottery website.

In Massachusetts, their lottery website says the overall odds of getting a multiplier of 3, 4, 5 or 10 is 1:2.3, this means that a multiplier of 1 occurs with a weight of 2.3 and the multipliers of 3, 4, 5 and 10 occur with weights of 0.25. So the average multiplier value in this case is

(2.3*1 + 0.25*(3+4+5+10))/(1+2.3) = 2.3636

Since it costs double the price of a ticket to get the multiplier bonus, the effective multiplier is 2.3636/2 = 1.1818. That seems like a more reasonable amount, because the game makers don't want the expected return to exceed \$1.

Also, you bring up a good point about the taxes!

• Jacquie 3 years ago

Your keno games in the States have better payouts than in Canada. I did you calculations and up here in BC they payouts are only about \$0.50 on the dollar for some n-spot games. We don't have taxes on lottery winnings though.

• Author

TR Smith 3 years ago from Eastern Europe

Interesting, Jacquie! Perhaps I should do an analysis of which states and provinces offer the greatest expected return for their keno games.

• Will 2 years ago

Hello C-G,

What if the game is played where the computer selects 30 random numbers from 1 to 90? Do the probabilities change or do they stay the same since you are adding 10 more numbers to the pool and 10 more numbers to the drawing?

• Author

TR Smith 2 years ago from Eastern Europe

Hi Will,

Thanks for your comment. The probabilities of winning various n-spot games do change if you add more numbers to the pool and drawing, even if you are adding the same amount. For example, in the 5-spot game with a pool of 90 and a drawing of 30, the probability of matching 5 out of 5 is (85 C 25)/(90 C 30) = 273/84194, or about 0.0032425. This is a lot better than the 80-20 5-spot game.

• Joe 9 months ago

What is the best 5 spot combination? Also there is a pattern to how the set combos that I caught rotate and reapear . Any the rooms on this?

• Author

TR Smith 9 months ago from Eastern Europe

Hi Joe, thanks for the question. I'm not sure what you're asking because you can't choose the combinations in regular vanilla keno that's just played on a computer. Are you asking about a lottery game or scratch ticket with 'keno' in it's name?