Lottery odds explained: how the jackpot probability is calculated, and what it means

Lotteries publish their odds, and the number is so large that it stops meaning anything: 1 in 45 million, 1 in 139 million. The maths behind it is one formula from school, and knowing it changes how the game looks. The lottery number generator picks fair random lines; this guide explains why the picking doesn't matter.

Counting combinations

Choosing 6 numbers from 49, in any order, can be done in "49 choose 6" ways: 49 × 48 × 47 × 46 × 45 × 44 ÷ (6 × 5 × 4 × 3 × 2 × 1) = 13,983,816. Exactly one of those combinations is drawn, so one line has a 1 in 13,983,816 chance of matching all six. Order doesn't matter (the balls are sorted after drawing), which is why we divide by 6!, and there is no replacement (a number can't be drawn twice), which is why the numerator counts down. Smaller prizes use the same counting: matching exactly 5 of 6 from 49 has 6 × 43 = 258 winning combinations, so 1 in 54,201.

The common formats

• 6 from 49 (the classic Lotto): 1 in 13.98 million.
• 6 from 59 (UK Lotto since 2015): 1 in 45.06 million.
• 5 from 50 plus 2 from 12 (EuroMillions): 2,118,760 × 66 = 1 in 139.8 million.
• 5 from 69 plus 1 from 26 (Powerball): 11,238,513 × 26 = 1 in 292.2 million.
• 5 from 70 plus 1 from 25 (Mega Millions): 1 in 302.6 million.

Bonus balls multiply the pool: the second machine's independent draw multiplies the main odds by its own combinations. Lotteries have added numbers and balls over the years precisely to lengthen the odds, which produces bigger rollovers, which sell more tickets.

Every ticket is equally unlikely

1-2-3-4-5-6 has exactly the same chance as any other line — it is one combination of 13.98 million — and the intuition that it "never comes up" is the same intuition that says the previous week's numbers are due or not due. Draws are independent; the machine has no memory (Is a coin flip really 50/50?). What choosing numbers can affect is the payout when you win: popular patterns (birthdays under 31, straight sequences, last week's numbers) are shared by many players, so a jackpot on those numbers is split more ways. A random line, or one avoiding dates and patterns, has the same chance of winning and a better chance of winning alone.

Expected value and rollovers

Expected value is the sum of each prize times its probability. For most draws it is far below the ticket price — around 50% is returned as prizes in many national lotteries, the rest going to good causes, tax and the operator — so the average ticket loses half its cost. Rollovers raise the jackpot without changing the odds, and at very large jackpots the expected value can approach or exceed the ticket price on paper; in practice the extra tickets sold raise the chance of sharing, and lump-sum and tax reductions cut the headline figure, so it rarely does. The percentage calculator and How to calculate percentages do the arithmetic; the honest summary is that a ticket buys a few days of imagining, and should be priced as entertainment.

Making the odds real

1 in 45 million: if you bought one ticket a week, you would expect a jackpot about once every 865,000 years. Filling a lottery machine's worth of tickets, at one a second, would take 17 months. The chance of being struck by lightning in a given year in the UK is around 1 in 10 million — four times more likely than the Lotto jackpot per ticket. None of that is a reason not to play; it is a reason to expect the £3 prize, enjoy the draw, and treat systems, hot numbers and "due" balls as fiction. The random picker and dice roller are the same randomness at friendlier odds; Dice probability explained: why 7 is the most common roll, and how to work out any odds the same counting on a smaller scale.