# Odds of Winning All or Nothing Lottery

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

All or Nothing is a simple lottery game that feels like a cross between keno and a traditional number draw game. To play All or Nothing, you select 12 numbers between 1 and 24 and see how well your set matches the lottery's set. If all of your numbers or none of your numbers match the numbers drawn by the lottery you win the top prize, hence the name of the game. This is because the probability of getting 12/12 correct is equal to the probability of getting 0/12 correct.

All or Nothing is played as a state lotto game in Texas, Arizona, North Carolina, and Georgia. In Iowa and Minnesota, it is played as a joint game, and the prize funds are pooled between the two states. In Arizona the pool of numbers is 1 to 20 and players select 10 instead of 12, which makes the probabilities different. Variations of the All or Nothing game were also offered for a short while in Ireland and Illinois.

Here are tables of the probabilities, followed by more detailed explanations of how they are calculated.

## Game Odds for Picking 12 Out of a Set of 24

Match
Odds to the Nearest Whole Number
12/12 and 0/12
1 in 2,704,165
11/12 and 1/12
1 in 18,779
10/12 and 2/12
1 in 621
9/12 and 3/12
1 in 56
8/12 and 4/12
1 in 11
7/12 and 5/12
1 in 4
6/12
1 in 3

## Game Odds for Picking 10 Out of Set of 20

Match
Odds to the Nearest Whole Number
10/10 and 0/10
1 in 184,756
9/10 and 1/10
1 in 1,848
8/10 and 2/10
1 in 91
7/10 and 3/10
1 in 13
6/10 and 4/10
1 in 4
5/10
1 in 3

## How Are These Figures Calculated?

Computation of lottery probabilities falls under a branch of mathematics called combinatorics. This article explains how to calcute them in the specific case of the All or Nothing game. To see how to calculate them in general, see also Basic Lottery Math and Advanced Lottery Math.

## Probability of Matching 12/12 or 0/12

The likelihood of getting 12/12 is equal to

1 / (24 choose 12)

where "(24 choose 12) is the combinatorial function. In other words, it's the number of ways of choosing 12 distinct objects out of a set of 24 distinct objects, without regard to the order in which they are drawn and without replacement. The probability is the reciprocal of that number. Using the combination function this gives you

1 / [24! / (12! * 12!) ]
= (12! * 12!) / 24!
= 1 / 2704156
= odds of 1 in 2,704,156
= 0.0000003698
= 0.00003698%.

When you pick 12 objects out of a set of 24, you are also leaving behind 12 objects. For every different set of 12 that can be picked, there is a unique disjoint set that is left behind. And for every possible set that can be left behind, there is a unique disjoint set that can be picked. This means the number of ways to pick all 12 correct is the same as the number of ways to pick them all incorrect. All is the same as nothing.

Since matching either 0/12 or 12/12 wins you the jackpot, and since each event has a probability of 1/2704156, the probability of winning the jackpot is actually 2/2704156 = 1/1352078. In other words the true odds of winning the jackpot in All or Nothing are about 1 in 1.35 million.

## Probability of Matching 11/12 or 1/12

Because matching exactly 12 or 0 is unlikely, All or Nothing lotto games usually offer a smaller prize for matching 11 out of 12 or 1 out of 12, the two events being equally likely by the same logic explained in the previous section.

The number of ways to get 11 numbers right and 1 wrong is equal to the number of ways of choosing 11 objects out of 12 "right" objects multiplied by the number of ways to choose 1 object out of of a set of 12 "wrong" objects. That number is

(12 choose 11) * (12 choose 1)
= 12 * 12
= 144.

The probability is 144 divided by the total number of different selections a player can make, which was calculated above as 2,704,156. Therefore, the probabilities of getting 11/12 and 1/12 are both equal to

144 / 2704156
= 0.00005325
= 0.005325%
= odds of 1 in 18778.8611.

By the same logic as in the previous section, the probability of winning the second prize by matching either 11/12 or 1/12 is 2*144/2704156, or approximately 0.0001065. This is equivalent to odds of 1 in 9389.4306.

## Probability of Matching 10/12 or 2/12

Most All or Nothing lotteries also have a third prize for matching 10/12 or 2/12, the probabilities of these two events also being equal. Continuing with the same methodology, the probability of each event is

(12 choose 10) * (12 choose 2) / (24 choose 12)
= 66*66 / 2704156
= 4356 / 2704156
= odds of 1 in 620.7888
= 0.001611
= 0.1611%.

The probability of winning the third prize is 2*4356/2704156 ≈ 0.003222, or odds of about 1 in 310.3944.

## All or Nothing Prize Payouts

The cost of a ticket and values of prizes vary from state to state. In Texas, Georgia, and North Carolina a ticket costs \$2 and players win

• \$250,000 for matching 12 or 0 out of 12
• \$500 for 11 or 1 out of 12
• \$50 for 10 or 2 out of 12
• \$10 for 9 or 3 out of 12
• \$2 for 8 or 4 out of 12
• nothing for 7, 6, or 5 out of 12

In the Iowa-Minnesota joint game a single play is only \$1 and players win

• \$100,000 for matching 12 or 0 out of 12
• \$1000 for 11 or 1 out of 12
• \$20 for 10 or 2 out of 12
• \$5 for 9 or 3 out of 12
• \$1 for 8 or 4 out of 12
• nothing for 7, 6, or 5 out of 12

The overall odds of winning anything (including a break-even prize) are about 1 in 4.54 for both lotteries. However, the expected returns are different. In Texas, Georgia, and North Carolina, players can expect to get back \$0.5598 on average for every dollar spent on All or Nothing games. In Iowa and Minnesota players get back \$0.6051 on average for every dollar spent. These returns are comparable to those of keno and much better than those of Powerball and Mega Millions. Of course, all lottery games are basically bad bets, and the best way to minimize your losses is to not play!

## Common Player Misconception Debunked

Some players avoid selecting sets featuring three or more consecutive numbers because they mistakenly believe such combos are unlikely to be drawn. However, three things that contradict this are

1. Any combination is as unlikely as any other.
2. You win by matching a set, not a pattern (or lack of pattern) in that set.
3. The chance that the lottery will pick a set of 12 numbers with at least three consecutive numbers is actually far greater than one-half. This is also true in Arizona where the master set of numbers is 1 to 20 and players select 10. There are comparatively very few combinations that manage to avoid three consecutive numbers, which is counter-intuitive to some players.

Any way you look at it, there's no valid mathematical reason for avoiding three consecutive numbers when you select your numbers.

12

0

2

2

0

## Popular

9

0

• ### Names of Geometric Shapes—With Pictures

30

0 of 8192 characters used

• Julio 14 months ago

I play this game in Texas because it's easier to win than Powerball and Mega Millions. But I never won more than \$10. If the numbers are drawn by a computer instead of a ball machine, is there a mathematical formula to predict what numbers are drawn?

• Judy Specht 14 months ago from California

You come up with the most interesting topics. When my eldest was 18 a friend bought him 18 scratchers. He would have rather of had the money than spent the time scratching to win \$3.

• Author

TR Smith 14 months ago from Eastern Europe

That sounds like a cute present. I know a woman who likes to collect scratch cards as souvenirs from all the different places she visits. Keeps them in mint condition, never scratched. Who knows if there's a winner among them?

• hugh 14 months ago

in arizona with 20 balls and a pick 10 system the odds of winning are 1 in 180k, but if you add 4 more balls and pick 2 more numbers (24 and 12) suddenly the odds jump to 1 in 2.7m? that doesn't make sense to me. if you increase the number of balls by 20% and the number of numbers you have to pick by 20% as well, shouldn't the odds be only about 40% worse? i admit i did not go further than pre-calculus and i saw the numbers you got are the same on the lottery website. what would the odds be if you had 28 balls and had to pick 14 numbers?

• Author

TR Smith 14 months ago from Eastern Europe

Hi Hugh, thanks for the question. The number of combinations does not grow linearly as you add more balls to the pool. If you created an All or Nothing game with the numbers 1 through 28 and players had to select 14 numbers out of that pool, the odds of matching 14/14 would be 1 in 40,116,600. This is because there are 40116600 ways to select 14 distinct objects from a set of of 28 distinct objects.

If you're wondering if these numbers grow according to a predictable pattern, the answer is yes. Say you start with 2N numbered balls (you always want an even number for these games and 2N is always even) and players choose N of these (N is half of 2N). Now suppose you add 2 more balls to the game so that the total number is 2N+2. Now players have to pick N+1 balls (N+1 is half of 2N+2). In this scenario the odds increase by a factor of

(4N+2)/(N+1)

This growth factor asymptotically approaches 4, meaning that when you add 2 more balls to the system (and ask players to choose 1 more number) the odds worsen by a factor of approximately 4. If you add 4 more balls the odds worsen by a factor of approximately 4*4 = 16. In other words, the odds grow approximately exponentially, rather than linearly.