# Should you buy 10 tickets for one lottery drawing, or play the lottery 10 times?

## Mathematically, what is the best way to distribute money used to buy lottery tickets?

Everyone knows intuitively that buying more tickets and playing more often increases the long-run odds of winning the lotto eventually. But if you only have $10 to spend on the lottery, will your odds of winning be better if you buy 10 different $1 tickets for a single drawing, or if you play a $1 ticket in 10 different drawings? And if one strategy is better than the other, by how much does the probability of winning increase?

The answers to these questions depend on the structure of the lottery, how you pick the numbers on your tickets, and what you mean by "winning" -- whether it is winning just the jackpot, or any prize amount. Here we explain how to compute these probabilities.

## "Winning" Defined as Winning Just the Jackpot

Let's take the Mega Millions lottery jackpot as an example. Recall that Mega Millions is played by picking five distinct numbers from a pool of 1 to 75, and one Mega Ball number from 1 to 15. The total number of different lottery ticket combinations for Mega Millions is 258,890,850, so the probability of hitting the lotto jackpot with a single ticket is 1/258890850 ≈ 0.00000000386263168. The probability of not hitting the jackpot is with a single ticket is 258890849/258890850 ≈ 0.99999999613736832.

If you buy 10 tickets for a single lottery drawing, and each ticket's numbers are disjoint from the any of the others' numbers, your probability of winning the jackpot jumps to

10/258890850

≈ **0.0000000386263168**

Now let's examine a different strategy to "invest" $10 in the lottery. If you buy one ticket for 10 different drawings, the probability that you'll win the jackpot at least once is

1 - (258890849/258890850)^10

≈ **0.0000000386263162**

Probability-wise, in terms of only winning the jackpot, the former strategy beats the latter strategy by about 0.0000000000000006. In practical terms, it makes virtually no difference which strategy you play.

## Related Lottery Math

More tutorials on combinatorics, statistics, and probability as applied to games of chance:

## "Winning" Defined as Winning Any Prize

Now let's look at what happens when you consider the probability of winning *any* lotto prize. The mathematics behind this comparison is more complicated. In the Mega Millions lottery, for example, the probability of winning something is 17602404/258890850 ≈ 0.0679916034. In other words, out of a possible 258,890,850 lottery combinations, 17,602,404 of them are eligible for some prize. The probability of not winning something is 241296446/258890850 ≈ 0.9320392976.

Suppose you purchase 10 different tickets for a single lottery drawing, and you select the numbers yourself so that no two tickets have the same set of main numbers (5 drawn from a pool of 75) and no two tickets have the same Mega Ball number (1 drawn out of a pool of 15). Then your probability of winning some prize could be estimated as 10*17602404/258890850 ≈ 0.6799160341.

However, this is not quite right, because any for two tickets with totally non-overlapping number selections, there are 49,002 ways that both tickets can win prizes (not the same prize though). Since there are 45 ways to make pairs out of 10 (since 10 choose 2 = 45), there are 45*49002 = 2205090 possible double-wins. (There is no way that three disjoint tickets can all win prizes.) These double-wins are double-counted in the previous computation. Therefore, we need to discount one copy of each double-win in order to calculate the exact probability of not losing. This means the probability of winning is actually

[10*17602404 - 45*49002]/258890850

≈ **0.6713985836**

Note that this calculation will not work when the number of tickets exceeds 15 in the Mega Millions lottery, since by the pigeonhole principle, some tickets will have to overlap in their numbers.

Now what if you purchase 10 tickets spread over 10 different drawings? In this scenario, your probability of winning any prize among the 10 lottery drawings is only

1 - (241296446/258890850)^10

≈ **0.5052996253**.

So with $10, it makes more sense probability-wise to spend all your money on a single lotto drawing rather than parcel it out over many lotto drawings. The former strategy gives you a 2/3 chance of winning, while in the latter gives you only a 1/2 chance of winning.

** ** ** ** ** ** **

You can perform the same calculation comparison for the Powerball or any other random number draw lottery. For example, in Powerball there are 175223510 possible ticket combinations, a 5502140/175223510 probability of winning something, and 29162 ways that two disjoint tickets can both win prizes. This means that the probability of winning with 10 tickets distributed over 10 different games is

1 - (169721370/175223510)^10 ≈ **0.2731554627**

and the probability of winning with 10 distinct tickets for a single drawing is

(10*5502140 - 45*29162)/175223510 ≈ **0.3065177156**

So it turns out that buying multiple tickets (with non-overlapping sets of numbers) for one drawing is a still a better strategy than entering one ticket at a time in multiple games.

## More Statistics and Probabilty

## Buying 1 Ticket to Cover Each Possible Red Ball Number

In the Mega Millions Lottery, there are 15 possible Mega Ball numbers. In the Powerball Lottery, there are 35 possible red ball numbers. In both lotteries, you win the lowest prize level if you match only the special ball or red ball number. Therefore, one strategy that increase your odds of winning to 100% is to buy one ticket for each possible special ball number.

For example, you could buy the following set of 15 Mega Millions tickets:

{1, 2, 3, 4, 5} + {1}

{6, 7, 8, 9, 10} + {2}

{11, 12, 13, 14, 15} + {3}

... {71, 72, 73, 74, 75} + {15}

Since every possible Mega Ball number is represented in this set of tickets, you are guaranteed to win at least the lowest prize level of $2. In fact, you also have a pretty good chance of winning the second lowest prize level for matching the Mega Ball plus one of the main numbers. Since a Mega Millions ticket only costs $1, your total outlay is $15.

For the Powerball lottery, there are 35 different special ball numbers, so you would have to buy 35 different tickets each with a different red ball number. Since a Powerball ticket costs $2, your total outlay is $70.

This lottery playing strategy ultimately results in a loss since since you end up spending more than you are guaranteed to make, and the expected return is still less than $1 per $1 wagered.

## It costs $70 to guarantee you'll win something in Powerball.

## Lottery Player Psychology

If it's better to play 10 tickets in a single drawing rather than one ticket apiece in 10 different drawings, why don't more lottery players employ this technique? There are millions of people who buy one ticket a week, every week of the year, for a total of 52 tickets per year. But there are far fewer people who buy 52 tickets for a single lottery drawing just once per year. Yet the math shows that the latter strategy greatly increases a players odds of winning a cash prize. What are these players thinking?

One reason is that playing in multiple games is more fun than playing only once. If people have a fixed amount to spend on the lottery, they tend to get more enjoyment out of playing the lottery as many times as possible. With the lump sum strategy, you blow all your money in one game and if you lose there's no more left to play with.

Another reason is that people can rationalize spending small amounts on long-shots, but it's harder to part with double-digit amounts of money for a sucker's bet like the lottery. However, people fail to realize that their small lottery ticket purchases here and there can add up to a lot of money over time.

## Bottom Line: Don't Play the Lottery as an "Investment"

Even if you can optimize your ticket buying with lump-buying so that the chance of winning something is greater than 50%, the expected return per $1 is always less than $1. Though you may win prizes more often than not, you end up paying more than the prizes are worth.

The lottery is a terrible investment and there are no mathematically-sound secret strategies for picking winning numbers or avoiding losing numbers, despite what the authors of strategy guides may assert. If they really knew the secret to winning, why would they tell? Only play for fun with the expectation that you are going to lose, and never spend more than you can afford to lose.

## Comments

An issue that reduces the "return on investment" of playing the lottery is the likelihood of sharing the big prize- even if you win the jackpot, there may be multiple winning tickets and you might end up with only 1/2 or 1/3 of the jackpot, etc. This is not such a bad problem if you win, but this reduces the potential value of each ticket. In the end, you are paying for entertainment when you buy lottery tickets...

I know now that in practical explanation, there is no right which can be sure of winning. But in your calculations, I learn how to make my choices. Thanks for sharing.

True story, for a while at our gas station we had an old lady come in once a month or so and employ this very strategy to buy Mega Millions tickets. Every time she would buy a set of 15 tickets each with different numbers. She knew at least one of them was bound to win at least the lowest prize. And sure enough the next week she would come in to claim her $1 prize for one of the tickets that had the right mega ball number. I guess there are worse ways to spend $14.

I've always thought this was a better strategy. Instead of playing once week by ourselves, everyone in our crew chips in a dollar to buy a bunch of tickets when the jackpot has rolled over a dozen times.

@KP, my lotto club used to do the same thing with PB and MM but now we switched to the lower stakes games. I'm glad I read this article because when our club first started out we were debating whether to chip in and lay down $50 on tickets once a month, or buy smaller amounts more frequently, and whether we should spread the funds out across different lotteries, or stick to a single game at a time. The math people in our group were saying to do it like you say, concentrate the money on a single game to maximize the chances of winning something, and to make sure all the different numbers are used on the payslips. I thought it was more annoying at first to do the payslips manually like that instead of quickpicks, but we divide up the labor of filling them out and we do usually win a small prize. the jackpot is still elusive.

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