ôô

# What Are the Odds of Winning the Lottery?

What are the odds of winning the Mega Millions lottery jackpot and striking it rich? How long would you need to play before having a 50-50 chance of winning? Keep on reading to find out!

By
Jason Marshall, PhD
Episode #213

Is buying a Mega Millions lottery ticket ever a "good" idea? If your goal is to have fun—and you find that sort of thing fun—then perhaps it is. But if your goal is to make a good investment, then a good idea it is not.

How high are the odds stacked against you? Can you improve your chances of winning by playing more often? And how many years, centuries, or millenia do you have to play to have a 50-50 chance of winning?

Stay tuned because those are exactly the questions we'll be talking about today.

.

## The Rules of Mega Millions

Simply put, the odds of winning the Mega Millions jackpot are not in your favor. As we'll see today, just how not in your favor they are is pretty astonishing. If nothing else, the very long odds we'll talk about serve as a nice reminder that if you do choose to play, you really should not expect to win the big prize. Ever.

Like most big-bucks lottery games, Mega Millions is a game where you select a bunch of numbers and hope they all match the numbers that show up on randomly drawn ping-pong balls. In order to calculate the actual probability of winning this game, we'll need to get a bit more specific with the rules. As described on the game's official website:

"Players may pick six numbers from two separate pools of numbers—five different numbers from 1 to 75 and one number from 1 to 15. You win the jackpot by matching all six winning numbers in a drawing."

Seems totally winnable, right? Looks can be deceiving. To get a better picture of your chances, let's calculate some probabilities.

## Probability and the Mega Millions Jackpot

In order to win the jackpot, you need to correctly guess five numbers between 1 and 75 from one pool as well as another number between 1 and 15 from a completely separate pool. What's the probability of guessing correctly?

Let's start by thinking about the five numbers drawn from the pool ranging between 1 and 75. How many sequences of five numbers from this pool are possible? Well, there are 75 numbers that can be drawn first. After drawing that initial number, there are 74 numbers that can be drawn second (since the numbers don't go back into the pool after being drawn). There are 73 possible numbers that can be drawn third, 72 that can be drawn fourth, and 71 that can be drawn fifth. This means that there are a total of

75 x 74 x 73 x 72 x 71 = 2,071,126,800

sequences of five balls that can be drawn from the first pool. But the folks at the lottery are generous and actually don't care about the order that you write down your guessed numbers. Since there are 5 x 4 x 3 x 2 x 1 = 120 different possible arrangements of the order in which those five balls were drawn (and since any of those sequences are counted as a match), there are actually a total of

2,071,126,800 / 120 = 17,259,390

possible order-independent combinations of five balls drawn from this pool.

But there's still that sixth ball from the second pool to worry about. Since there are 15 possible numbers in that pool, the total number of possible combinations of all six balls in the Mega Millions lottery game is:

17,259,390 x 15 = 258,890,850

And that is a really big number. Which is bad news for hopeful players because the chances of winning are 1 over this number. In other words, the probability of winning the Mega Millions jackpot is 1 in 258,890,850 or about 0.0000004%—not so good.

## How to Get a 50-50 Chance of Winning

But all hope is not lost because there is a way to improve your chances of winning. Well, sort of. The secret is to play for a long, long—really, really long—time. Did I mention it has to be a long time?

To see what I mean, let's think about how long you would have to play in order to have a 50-50 chance of winning. As we learned when we solved the birthday problem, the easiest way to solve problems like this is often to think not about the probability of winning after a number of attempts, but instead to think about the probability of not winning—i.e., losing—in each and every one of those attempts, and then to subtract this probability of losing from 1 to find the probability of winning.

There's a 50-50 chance that you'll win the jackpot sometime in the next 1.7 million years!

Every time you play the Mega Millions, the probability of losing is 258,890,849/258,890,850—which is a 99.99999 (and then some more 9s) percent chance of losing. The probability of losing twice in a row is this number squared, the probability of losing 100 times in a row is this number raised to the 100th power, and the probability of losing N times in a row is this number raised to the Nth power.

If you calculate (258,890,849/258,890,850)180,000,000, you'll find that it's approximately equal to 50%. Which also means that your odds of winning the Mega Millions jackpot finally reach the 50-50 mark after playing 180 million times! Since the game is played twice a week, you could play 104 times per year. If you buy 1 ticket each game, there's a 50-50 chance that you'll win the jackpot sometime in the next 1.7 million years!

As I said, the odds of this game are not in your favor. Of course, odds are just odds—and occasionally anomalies happen. But I wouldn't bet on it.

## Wrap Up

OK, that's all the math we have time for today!

Please be sure to check out my book, "The Math Dude’s Quick and Dirty Guide to Algebra." And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Lottery balls image courtesy of Shutterstock.