Life-changing money for only a $2 ticket feels like a dream. And many Americans chase that dream every week. In fact, in 2020 Americans spent$90 billion on the lottery, particularly on tickets for the Powerball and Mega Millions jackpots. To raise the stakes, lotteries have become more and more difficult over time, with greater numbers of losing tickets funding ever-larger jackpots. This lure of massive jackpots has led to more players buying tickets — and more profits for states.

But if you do decide to play the lottery, what are your actual chances of winning money? And what about BIG jackpot money?

In this article, we'll dive into the math behind lotteries to calculate the probability of winning the jackpot — and some strategies to help increase those chances.

## Starting Small: How Did Lotteries Raise the Stakes?

Lotteries in the U.S. have taken many forms over the years. State-sponsored lotteries began with ticket-based raffles in the 1960s. Originally, the probability of winning the raffle depended on the number of tickets sold.

Then, lotteries evolved to daily number drawings, starting in New Jersey in May 1975. The probability of winning this type of lottery depended on how often you played. In these drawings, players guess at a 3- or 4-digit number drawn at random.

For example, if you pick the perfect number 8128, you have a 1 in 10 chance of getting the first digit right, another 1 in 10 for the second, and so on. Ultimately this leads to a 1 in 10,000 chance of winning.

Another way to play is to guess that the digits 8128 come out in any order. Normally there would be 4! (pronounced "four factorial" and equal to 4 x 3 x 2 x 1 = 24) ways for a four-digit number to be ordered, but the repeated 8 gives 4!/2!, or 12, possibilities out of the 10,000 in the game. So the chance of winning this way is about 1 in 833. While the odds are better, the prizes tended to be lower as a result.

### The Probability of Winning Pick-6

Soon, Pick-6 games emerged in popularity. Balls with the numbers from 1 to 30 bounced around in a machine, you pick six numbers, then six balls are drawn. If you match 3, 4, or 5 numbers, you win a small prize. If you match all six numbers, you win the jackpot.

But how likely are you to win the Pick-6 jackpot?

Let's look at combinations. There's only one winning combination, but how many combinations are there in total? The lottery drawing divides the 30 numbers into two categories: the 6 winning numbers and the 24 losing numbers. The order does not matter within these categories, so we can use the same logic we used when looking at 8128.

The number of combinations is

$\frac{30!}{6! • 24!} = 593,775.$

Imagine the lottery machine dumping out all 30 balls in order. The order of the first six numbers is not important, just which numbers are picked. The similar principle applies to the last 24 balls: The order does not matter because they all lose. You can see this within the equation above: There are 30! ways for the lottery machine to dump all 30 balls, and 30! is divided by the number of different orderings that do not matter.

#### The odds of winning the Pick-6 jackpot is 1 in 593,775. That’s really rare!

But it’s far less rare than trying to call the numbers in the exact order. Part of the appeal of Pick-6 lotteries is that any of your numbers could be drawn at any time; even if you miss a couple of numbers, you could still win something.

These calculations are common in combinatorics and probability. The phrasing is “30 choose 6,” which describes exactly what is done: choosing 6 things out of 30, regardless of order. This notation represents that same phrasing:

${30 \choose 6}.$

This number can also be found in the 30th row of Pascal’s Triangle.

We can use the same concept to determine how many ways we can get 4 numbers right. We must pick 4 of the 6 correct numbers, and 2 of the 24 incorrect numbers. The number of outcomes where there are exactly 4 of 6 correct numbers is

${6 \choose 4}• {24 \choose 2} = 4,140.$

Therefore the chance of getting exactly 4 out of 6 numbers correct is 4,140/593,775, nearly 1%. Similar work determines all possible outcomes and probabilities, including the fact that almost 80% of tickets in this lottery will get at least one number right.

### What Happens If No One Picks the Correct Set of Pick-6 Numbers?

When no one wins a jackpot, it increases for the next drawing. This encourages more people to play, hoping to hit the bigger jackpot.

Lottery organizers noticed that more people played when the jackpots were larger. Many U.S. states and other governments started to increase the number of balls in the lottery in the hopes of bringing in even more players. The number of balls typically changed to fit grids that players used to fill out their entries: 35 balls, 36 balls, 42 balls.

In 1982, Canada started running Lotto 6/49. At first glance, it doesn’t seem like picking 6 out of 49 should be that much more difficult than picking 6 out of 30, but it is over 20 times harder:

${49 \choose 6} = \frac{49}{6!•43!} = 13,983,816.$

Lotto 6/49 was a huge success. In early 1984, Lotto 6/49 offered an unprecedented jackpot of over $13 million CAD (Canadian dollars, roughly$10 million US dollars) with over 60% of all Canadian adults participating.

The jackpot was big, but the money raised for the government through ticket sales was bigger.

## Introducing the Powerball: Pick-5+1

In 1988, several U.S. states got together to build their own version of Lotto 6/49. The game was a Pick-7, with 40 numbers. Though the odds were about the same as Lotto 6/49, about 1 in 18 million, it didn't do well. Players felt that picking 7 numbers was too difficult.

Enter Powerball. Starting in 1992, players still picked six numbers from 1 to 45, but the sixth number, the Powerball, is drawn from a different lottery machine. That is, the five white balls are drawn from one machine with balls numbered 1–45, and then a bonus ball is drawn from a second machine holding another set of balls numbered 1–45.

Can you figure out what this does to the number of outcomes?

### Chances of Winning the Powerball

Powerball is called a Pick-5+1 lottery, rather than a Pick-6. The bonus ball must be picked precisely. It can even be the same number as one of the white balls! All of this makes winning a jackpot far more difficult, while still keeping the appearance of a Pick-6. You could win a small prize in Powerball just for hitting the bonus ball, and since the Powerball is drawn last, any player always has a chance to win until the drawing is complete.

To win the jackpot, a player must correctly choose the 5 winning numbers among the 45 white balls, and precisely call the winning number on the bonus ball. The number of outcomes of the original, 1992 version of Powerball is

${45 \choose 5} • 45 = \frac{45!• 45}{5!• 40!} = 54,979,155.$

#### Each ticket had only a 1 in 55 million chance of winning in an original 1992 Powerball drawing.

To put this in perspective, you are less likely to hit this jackpot than to flip a coin and have it come up heads 25 times in a row, because 2^(25) is about 33.5 million.

## Big-Money Lotteries Get Even Bigger — and Harder to Win

Powerball was a huge success for the states that offered it: those states generated more revenue from lottery tickets. More states joined, and other states built their own Pick-5+1 lottery, now called Mega Millions.

The two groups eventually merged. Today, all 45 states that sell lottery tickets sell both Powerball and Mega Millions. (The states that don’t offer a lottery are Alabama, Alaska, Hawaii, Nevada, and Utah.)

Every few years, Powerball and Mega Millions make small adjustments to the numbers of balls, but almost always in ways that make the jackpot more difficult to win. Even small increases in the number of white balls make a huge impact. These are some of the options Powerball and Mega Millions have used.

Today, Mega Millions uses 70 white balls and 25 bonus balls, whereas Powerball uses 69 white balls and 26 bonus balls. Powerball has 292,201,338 outcomes, making it slightly easier to win than Mega Millions. For both, we can reasonably use 300 million as an estimate for the number of outcomes.

#### Each ticket has only a 1 in 292 million chance of winning in a 2022 Powerball drawing.

Because of these changes, winning Powerball or Mega Millions is now more difficult than flipping a coin and having it come up heads 28 times in a row. While 28 times in a row doesn’t sound that much more difficult than 25, imagine flipping heads 25 times in a row and knowing that you still have only a 1 in 8 chance of winning!

Additional lottery changes included increasing the price of tickets from $1 to$2. And drawings are now more frequent: Each week has 3 Powerball and 2 Mega Millions drawings, instead of one weekly drawing.

All of this creates more massive jackpots. In January 2022, the Powerball jackpot passed $500 million after dozens of consecutive drawings with no winner. In April 2022, the Powerball jackpot was back over$300 million again. This isn't unusual: Powerball and Mega Millions have reached a $300 million jackpot over 50 times, and a$500 million jackpot 20 times.

Remember: Every dollar of those jackpots is being funded by other players’ losing tickets. One dollar from each ticket goes toward prizes, with the other as revenue. Almost all of the prize money goes into the jackpot, so the jackpot value is a rough estimate of the number of losing tickets over multiple drawings. When the jackpot is at $300 million, there have been roughly 300 million losing tickets bought since the last jackpot winner. ## But There Has to Be a Winner – and That Winner Could Be Me! There has to be a winner in 1 out of 300 million tickets, right? It's less likely than you think. ### Chances of Winning a$300 Million Jackpot

The chance of buying a ticket and not winning the jackpot is 1 - 1/N, where N is the total number of outcomes (302,575,350 for Mega Millions). If there are T tickets purchased, the chances of none of them winning the jackpot can be estimated as

$(1-1/N)^T$

using an assumption that each ticket is independent of all others.

When we set N = 302,575,350 and T = 300,000,000, we get a probability of 37.1%, or roughly 1 in 2.7.

Just change T to 1,000,000,000. The probability of a billion losing tickets is about 3.7%, or roughly 1 in 27.

Let’s check this against the real data.

As of February 2022, there have been 67 jackpots paid by Powerball and Mega Millions with their current formats. 27 of 67 (40.3%, 1 in 2.5) have been for at least $300 million, and 3 of 67 (4.5%, 1 in 22) have been for at least$1 billion. The math is very accurate!

There is also a connection to the important mathematical constant known as e. The value of e can be defined as the limit of $(1+1/N)^N$ as N grows larger and larger, or about 2.718. This definition also means that 1/e is the limit of $(1-1/N)^N.$Letting N and T be 300 million in our earlier formula tells us that 1/e is roughly the probability of seeing 300 million consecutive losing Mega Millions tickets. Letting T be 600 million tells us that there is roughly a 1/(e^2)(1 in 7.4) chance of seeing 600 million consecutive losing tickets, estimating the chance that a Mega Millions jackpot makes it all the way to $600 million before anyone wins. Similarly, we can estimate a 1 in 20 chance that a jackpot will cross$900 million before anyone wins, and a 1 in 400 chance that it will cross $1.8 billion. ### What If I Bought All the Tickets? Can playing Powerball or Mega Millions lotteries ever be profitable? To figure this out, think about what would happen if we bought all 302,575,350 Mega Millions tickets. Buying all the tickets wins a guaranteed jackpot, along with every other smaller prize in the lottery. We can compute the number of times each small prize is won and the total amount that will be won (besides the jackpot). Mega Millions has 70 white balls and 25 bonus balls, so the calculations split the 70 white balls into 5 winning balls and 65 losing balls. The good news about this strategy? If you buy all the tickets, you’ll win almost$75 million, plus the jackpot!

The bad news about this strategy? It costs $605 million to buy all the tickets. So is buying all the lottery tickets still profitable? It might look like if the jackpot is greater than about$530 million, then this would be profitable! But this is not true, for two reasons:

• The jackpot is paid in installments over 20 years. Most winners opt to take a smaller, immediate payout. The actual payout is typically about half the advertised jackpot.
• If two or more people each win, the jackpot is split evenly. When the jackpot is more than $500 million, this happens roughly one-third of the time. And you’d better hope nobody else also tries to buy all the tickets! In the history of lotteries, a single ticket has been worth over$530 million only three times. And two of those tickets were won in 2021.

If you’re wondering, buying every ticket has been done many times in smaller lotteries, both in the U.S. and Europe. Some lotteries give out bigger prizes when their jackpots grow too high, and this group from Random Hall at MIT won big.

## Still Want to Play? Strategies For Playing to Win Big

The only way a player can increase their odds of winning the jackpot is to buy more tickets. However, there is a strategy:

Choose your numbers in ways that decrease the chances of splitting the jackpot

Tip #1

Lottery players use the numbers 1–31 far more often than numbers larger than 31, due to their association with dates, according to an analysis in Judgment and Decision Making. Lottery players also play numbers and combinations with a visual or cultural appeal, such as marking a full column, or playing numbers found in a fortune cookie or the TV show Lost. Such choices can reduce a ticket’s potential to win big by making it more likely to split any jackpot winnings.

While there is no single set of best numbers, the best advice for anyone picking lottery numbers is to stay completely away from all of 1 through 31, because they’re played more than any other numbers. It’s also a good idea to keep clear of numbers like 42 that have pop culture recognition.

Tip #2

Typical lottery players also tend to space out their numbers, so players seeking to make a unique combination may want to pick at least two numbers consecutively or near-consecutively.

Players who pick a number within 1–25 should also pick the same number for the bonus ball, as few players mark their bonus ball to be the same as one of the five white balls.

Tip #3

Lastly, players should generally avoid playing a “booster” bet that increases the value of non-jackpot payouts. The non-jackpot payouts are low, relative to the jackpot. Unless the jackpot is fairly small (and by “fairly small,” we mean around \$200 million or smaller), the booster does not increase the small payouts enough to be worthwhile. With a large jackpot, money spent on booster bets would be better spent on additional tickets that increase the chances of hitting a jackpot.

Like many of the other changes in the 40+ years of big lotteries in the U.S., many of the lottery decisions were made to maximize the number of players and tickets. Powerball and Mega Millions are built to give players just enough to go on: Pick six numbers and win ridiculous money!

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Do big jackpots increase your chances of playing, even though they decrease your chances of winning? What would you do if you ran the lottery? Tag us at @artofproblemsolving with #mathoflotteries.