How lottery odds are calculated

Lottery odds get quoted constantly and explained almost never. They aren't guesswork, though, and they aren't a secret: every figure an operator publishes comes out of one piece of school-level maths you can check yourself in about a minute. Below is that formula, worked through slowly, then applied to real games to show it reproduces the official numbers exactly.

The one formula you need

A lottery draw picks a handful of numbers from a larger pool, and — crucially — the order doesn't matter. Drawing 7, 12, 30 is the same result as drawing 30, 7, 12. That single fact is what the formula is built around.

Start with the easier question: how many ways can you draw 5 numbers from 50 if order does matter? The first ball can be any of 50, the second any of the 49 left, then 48, 47, 46:

50 × 49 × 48 × 47 × 46 = 254,251,200

But that counts every set many times over — once for each order it could have come out in. Five numbers can be arranged in 5 × 4 × 3 × 2 × 1 = 120 different orders, so divide that out:

254,251,200 ÷ 120 = 2,118,760

There are 2,118,760 possible sets of 5 numbers from 50. That's the whole idea. Written formally it's the combinations formula, usually said aloud as "50 choose 5":

C(n, k) = n! ÷ (k! × (n − k)!)

Your ticket is one of those sets, and in a fair draw each is equally likely — so your chance of matching all five main numbers in a 5-from-50 game is exactly 1 in 2,118,760.

A neat check on that figure: 1 in 2,118,760 is precisely the published odds of the top prize in EuroMillions HotPicks Pick 5, which asks you to match five main numbers from 1–50 and nothing else. The maths and the operator agree.

Two pools multiply

Most big games draw from two separate machines — main numbers from one, a bonus ball from another. Because the second draw is independent of the first, you multiply the two counts.

EuroMillions asks for 5 numbers from 50, plus 2 Lucky Stars from 12:

• Main numbers: C(50, 5) = 2,118,760
• Lucky Stars: C(12, 2) = 66
• Together: 2,118,760 × 66 = 139,838,160

That is the published jackpot odds for EuroMillions, to the digit. Notice how much work those two extra star balls do — 66 times more combinations, from a pool of just twelve.

Every game, checked

Same method, eight formats. The right-hand column is the operator's own published figure, and in every case it's exactly what the arithmetic produces.

Two things jump out. Powerball and Mega Millions are in a different league — roughly twice as long as EuroMillions, and around thirty-six times longer than Thunderball. And EuroMillions and EuroJackpot have identical jackpot odds, because they use the identical 5-from-50 plus 2-from-12 structure. Their prize funds and jackpot caps differ enormously; the underlying draw does not.

The HotPicks twist

HotPicks games look like they should follow the same rule, and they don't. It's an easy trap — I nearly fell into it writing this page.

In Lotto HotPicks you might pick just 3 numbers from 59, but your 3 have to appear among the 6 main numbers drawn. You get six numbers' worth of chances to be right, not one.

So the count of favourable outcomes isn't 1, it's C(6, 3) = 20 — the number of 3-number sets hiding inside the 6 drawn:

C(6, 3) ÷ C(59, 3) = 20 ÷ 32,509 = 1 in 1,625

Treat it as 1 in C(59, 3) instead and you'd overstate the difficulty twentyfold. The general form is C(drawn, k) ÷ C(pool, k).

Why Lotto's odds nearly halved in June 2026

On 7 June 2026 Lotto moved to a two-round format: one line, entered automatically into two separate draws. Your chance of winning something roughly doubles — but not exactly, and the reason is worth a moment.

If p is your chance in a single round, the chance of winning in at least one of two rounds isn't 2p. That double-counts the occasions you'd win in both. The correct expression subtracts the overlap:

2p − p²

Matching all 6 in a single Lotto round is 1 in 45,057,474. Across two rounds it becomes 1 in 22,528,738 — a shade worse than half, because of the vanishingly small chance of hitting it twice. At the low-value tiers the effect is easier to see: Match 3 lands at 1 in 48.4, not the 1 in 48.1 naive halving would give. The published figures use the correct maths.

What these numbers actually feel like

Enormous odds stop meaning anything past a point, so here they are as time instead. Playing one line in every single draw:

• Thunderball (4 draws a week) — about 38,000 years for the top prize.
• Set For Life (2 a week) — about 147,000 years.
• Lotto (2 a week) — about 217,000 years.
• EuroMillions (2 a week) — about 1.3 million years.
• Powerball (3 a week) — about 1.9 million years.

Another angle: buying every possible EuroMillions line once, at £2.50 each, would cost roughly £350 million. Which is why "just buy them all" only works in heist films, and only for jackpots bigger than that.

Why "1 in 13" is true and slightly misleading

EuroMillions advertises overall odds of 1 in 13 of winning a prize. That's accurate. It's also mostly counting the bottom tier: match 2 main numbers and you win £2.50, which is exactly your stake back. You haven't won anything — you've had your money returned.

The same holds across the board. The great bulk of "wins" are small fixed prizes hovering near the ticket price. Overall-odds figures describe how often something happens, not how often you come out ahead.

On which: these games are designed to pay out roughly half of what they take in. Lotto and Set For Life run near 50–52%, Thunderball 52%, EuroJackpot 50%. That missing half is the point — it funds prizes, the operator, retailers, duty and good causes. No combination of numbers, no system and no generator changes it, because the shortfall is structural rather than something clever play can dodge.

So can anything improve your chances?

Only buying more lines, and only in proportion to what you spend. Two lines double a microscopic chance and cost twice as much; the expected return per pound doesn't budge.

What number choice can affect is who you share with. Plenty of people pick birthdays, so lines confined to 1–31 tend to be more heavily duplicated, and a jackpot landing on one gets split more ways. Avoiding those patterns doesn't make you likelier to win — it just means a larger slice if you do. That's the honest case for a random pick, and it's covered properly in do lucky numbers work?.

Check the maths yourself

Every figure here came from the combinations formula and was verified against the operators' published odds. If you want to confirm any of it, most scientific calculators have an nCr button, and in a spreadsheet it's =COMBIN(50,5). Nothing on this page requires taking my word for it, which is rather the point.

Read next

• Do lucky numbers work? →
• EuroMillions guide →
• Lotto guide →
• Responsible gambling →

Lucky Dip is for entertainment, and for over-18s. If gambling has stopped being fun, free and confidential help is available.