Keno Lottery Strategies, Odds, & Expected Winnings
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 nspot game he would like to play. Each nspot game can be played for a $1 wager. In the 1spot game a player chooses 1 number from 1 to 80; in the 2spot game he chooses two different numbers from 1 to 80; in the 3spot 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 1spot, 2spot, ..., or 10spot 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 nspot 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 nspot 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 10spot 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 nspot 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)*(80n C 20k)]/(80 C 20)
where (x C y) is the combinatorial function "x choose y." For example, if you play the 5spot 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
Calculating Overall Chance of Winning (or not losing)
To figure the overall chance of winning a prize for an nspot game, including any breakeven $1 prizes, you simply add up all the probabilities of winning for the all the koutofn matches. When breakeven prizes are included in the tally, you are technically calculating the odds of not losing, to be more precise. Let's use the 5spot game as an example again.
We already saw in the previous section that the probability of matching 3 out of 5 in the 5spot 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 5spot 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 nspot games for every value of n from 1 to 12. This produces the following table of overall odds for each nspot game in the Massachusetts keno lottery.
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 2spot game, followed by the 4spot game. The game with the worst overall chance of not losing is the 12spot game, followed by the 5spot game. One might think then that the 2 and 4spot games are the least risky, while the 12 and 5spot 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 nspot 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 koutofn probability by the payout for that match, then summing those numbers. Let's use the 5spot 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' 5spot keno game.
The table below summarizes the expected earnings for every nspot game in Massachusetts keno lottery.
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 5spot game offers the highest return on your wager, while the 1spot game offers the least. Therefore, to maximize your earnings, or minimize your losses, you should play the 5spot game. The differences in the table may seem negligible, but in aggregate if you take all the people who play the 5spot game and all the people who play the 1spot game, those who play the 5spot 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 nspot game.
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Comments 10 comments
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.
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 5spot 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.
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 nspot games. We don't have taxes on lottery winnings though.
Hello CG,
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?
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?
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