What Are the Odds of Winning the Lottery? The Math Explained in Plain English
Everyone knows the lottery is a long shot. Far fewer people know how long, or why. The odds printed on the back of a ticket aren't a guess or a marketing number. They come from a single, simple piece of math you can do yourself, and once you understand it, you'll never look at a jackpot headline the same way.
Every Combination Is Equally Unlikely
Most players believe some number combinations are "more likely" than others. A ticket with 1-2-3-4-5 feels impossible. A ticket with 7-19-23-41-58 feels reasonable.
In a fair draw, both have exactly the same chance. The draw machine doesn't know which numbers look random to humans. What makes 7-19-23-41-58 feel likely is that there are millions of "random-looking" combinations and only a handful of "patterned" ones, so random-looking results come up far more often as a group. Any single combination, patterned or not, is equally rare.
We call this the pattern illusion: mistaking how common a type of result is for how likely a specific result is. It's the root of a lot of lottery myths.
The Core Idea: Counting Combinations
Lottery odds come from a branch of math called combinatorics — specifically, counting how many different ways you can choose a set of numbers when order doesn't matter.
The formula is called "n choose k," written as C(n, k):
C(n, k) = n! / (k! × (n − k)!)
Where:
- n is how many numbers are in the pool
- k is how many numbers get drawn
- ! means factorial (5! = 5 × 4 × 3 × 2 × 1)
You don't need to love factorials. You just need to know that this formula counts every possible ticket.
Example 1: A Simple 6/49 Lottery
Canada's Lotto 6/49 draws 6 numbers from 1 to 49. How many different tickets are possible?
C(49, 6) = 13,983,816
So your odds of matching all six numbers with one ticket are 1 in 13,983,816. If you bought one ticket per draw, twice a week, you'd expect to wait over 130,000 years for a jackpot on average.
Try the Canada Lotto 6/49 number generator.
Example 2: Powerball
Powerball uses two separate pools:
- 5 white balls from 1 to 69
- 1 red Powerball from 1 to 26
First, count the white-ball combinations:
C(69, 5) = 11,238,513
Then multiply by the 26 possible Powerballs:
11,238,513 × 26 = 292,201,338
That's why Powerball's jackpot odds are 1 in 292,201,338.
Try the Powerball number generator.
Example 3: Mega Millions
Since April 2025, Mega Millions has drawn 5 white balls from 1 to 70 and 1 gold Mega Ball from 1 to 24:
C(70, 5) × 24 = 12,103,014 × 24 = 290,472,336
Jackpot odds: 1 in 290,472,336.
Try the Mega Millions number generator.
How Big Is 1 in 292 Million, Really?
Big numbers stop meaning anything after a few zeros. Here are a few ways to picture them:
- If you bought one Powerball ticket for every draw — three draws a week — you'd expect to wait roughly 1.87 million years for a jackpot.
- If every Powerball combination were printed on a 4-inch ticket and laid end to end, the line would stretch more than 18,000 miles — roughly three-quarters of the way around the Earth.
- You would need to buy about 146 million tickets to have a coin flip's chance (50%) of winning a single draw.
We call this the zero fog: the way very large and very small probabilities blur together in human intuition. "1 in a million" and "1 in 300 million" feel similar, but one is 300 times less likely.
Why the Smaller Prizes Are Far More Likely
Jackpot odds get the headlines, but lotteries pay many lower prize tiers. Powerball's overall odds of winning any prize are about 1 in 24.87, mostly from small prizes like matching just the Powerball.
The math for a lower tier works the same way: count the combinations that match exactly the right amount. For example, matching 5 white balls but missing the Powerball happens in 25 of the 292,201,338 combinations (the one right white-ball set with any of 25 wrong Powerballs), giving odds of about 1 in 11.7 million.
That's still rare, but it's 25 times more likely than the jackpot.
Comparing Popular Lotteries
| Lottery | Format | Jackpot odds |
|---|---|---|
| Canada Lotto 6/49 | 6 of 49 | 1 in 13,983,816 |
| UK Lotto | 6 of 59 | 1 in 45,057,474 |
| EuroMillions | 5 of 50 + 2 of 12 | 1 in 139,838,160 |
| Mega Millions | 5 of 70 + 1 of 24 | 1 in 290,472,336 |
| Powerball | 5 of 69 + 1 of 26 | 1 in 292,201,338 |
| Italy SuperEnalotto | 6 of 90 | 1 in 622,614,630 |
Notice the pattern: adding more balls to the pool doesn't increase odds a little — it multiplies them. Going from 49 to 59 numbers in a 6-number game makes the jackpot more than three times harder.
Why Lotteries Make the Odds Harder on Purpose
Lotteries tune their odds to create huge jackpots. Longer odds mean fewer jackpot winners, which means more rollovers and bigger advertised prizes, which sells more tickets.
Powerball changed its format in October 2015, moving from 59 white balls to 69 and reducing the red balls from 35 to 26. The jackpot odds got longer, but the overall odds of winning something improved — and record-breaking jackpots followed within months.
The economist and Nobel laureate Daniel Kahneman, whose work with Amos Tversky on prospect theory explained how people misjudge probabilities, showed that we tend to overweight very small chances. Lottery designers don't need to hide the odds. Our brains already struggle to feel them.
Quick Answers
What are the odds of winning the Powerball jackpot?
The odds of winning the Powerball jackpot are 1 in 292,201,338. That number comes from multiplying the 11,238,513 ways to choose 5 white balls from 69 by the 26 possible red Powerballs. The odds of winning any Powerball prize are much better, at about 1 in 24.87.
Does buying more tickets improve your odds?
Yes, but only in a linear way. Two different tickets double your chance, and ten tickets make it ten times better. For Powerball, ten tickets still leaves you at about 1 in 29 million. Buying multiple tickets with the same numbers doesn't help your odds at all.
How do I calculate lottery odds myself?
Use the combination formula C(n, k) = n! / (k! × (n − k)!), where n is the size of the number pool and k is how many numbers are drawn. If the lottery has a separate bonus ball pool, multiply the result by the number of possible bonus balls.
Where Lottery Odds Are Heading
Jackpot sizes have been climbing for decades, and lotteries keep adjusting their formats to push them higher. Our prediction: as record jackpots become more common, more lotteries will follow the Powerball and Mega Millions approach of longer jackpot odds paired with better odds on small prizes, keeping players engaged between life-changing jackpots.
For how to pick numbers once you know the odds, read How to Choose Lottery Numbers (Smart vs Random).