Lottery Math

Set n, k, and prize amounts. The tool computes all probabilities from C(n,k), expected value per tier, and your lifetime jackpot odds.

Worked example — Lottery Math

Years to 50% chance of jackpot	1.9M years
Jackpot odds	1 in 139,838,160
Total combinations	139,838,160
Return rate	28.6%
EV / ticket	€-0.857
Break-even jackpot	€134.9M
Return to players	28.6%
Operator margin	71.4%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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expected value and the 292-million-year problem 🖖

Powerball's jackpot odds are 1 in 292,201,338 — match 5 numbers from 69, then the Powerball from a separate drum of 26; C(69,5) × 26 gives that exact figure. At one ticket a week, the geometric distribution says you'd wait roughly 5.6 million years on average for a single jackpot win. Buying more tickets barely moves the needle: 10 tickets a week just cuts the wait to 560,000 years, because the odds scale linearly while the expected value per ticket stays negative — the prize pool never fully returns what was collected in ticket sales. The lottery isn't a bad investment; it's not an investment at all.

two lotteries, different rules, identical odds 🖖

At its heart this tool just counts: every valid ticket is one combination, all of them equally likely, so the odds are one divided by the count. What that hides is how differently two games can reach the same count. The German 6/49 draws six balls from 49 and adds a Superzahl from ten, giving 13,983,816 × 10. EuroMillions draws five from 50 and adds two Lucky Stars from twelve, giving 2,118,760 × 66. Both come out at exactly 139,838,160 possible tickets — not approximately, exactly, because C(49,6) is 6.6 times C(50,5) and 66 is 6.6 times 10. Two committees in two countries, and the same number of ways to lose.

the man who bought a lottery 🖖

In 1992 a syndicate led by mathematician Stefan Mandel targeted the Virginia Lottery, which had only C(44,6) = 7,059,052 possible tickets at $1 each, while the jackpot had rolled over to about $27 million. When the prize pool exceeds the cost of buying every combination, expected value flips positive — so they set out to purchase them all. They managed to print roughly 5 million tickets in time, and won. Using similar brute-force combinatorics, Mandel won lotteries 14 times.

Six lotteries, one formula

Powerball’s main draw is easier than the German 6/49’s

Matching five numbers from 69 is easier than matching six from 49 — 1 in 11,238,513 against 1 in 13,983,816. Every bit of Powerball’s extra difficulty comes from the second drum. That is what the split below shows: a jackpot is the main draw multiplied by the bonus, and designers trade the two against each other.

Lottery	The draw	Main draw, 1 in	Bonus factor	Jackpot, 1 in
UK Lotto	6/59	45,057,474	—	45,057,474
Viking Lotto	6/48 + 1/8	12,271,512	×8	98,172,096
German Lotto 6/49	6/49 + 1/10	13,983,816	×10	139,838,160
EuroMillions	5/50 + 2/12	2,118,760	×66	139,838,160
EuroJackpot	5/50 + 2/12	2,118,760	×66	139,838,160
US Powerball	5/69 + 1/26	11,238,513	×26	292,201,338

Read the middle two columns against each other. EuroMillions has by far the easiest main draw of the six — 1 in 2,118,760, twenty times easier than the UK’s — and still ends up among the hardest jackpots, because its two Lucky Stars multiply the count by 66. The UK is the only game here with no second drum at all, which is exactly why its jackpot is the most reachable. And three of the six tie to the digit at 139,838,160: 6/49 and EuroMillions arrive there from opposite directions, and EuroJackpot has matched EuroMillions’ structure since 2022. The identity is exact rather than a near miss — C(49,6) is 6.6 times C(50,5), and 66 is 6.6 times 10. Every figure in the table is computed from the same C(n,k) the tool runs, and every jackpot column is checked against that game’s published odds at build time.

References (5)

UK Lotto prize-tier odds — jackpot 1 in 45,057,474, five plus bonus 1 in 7,509,579: Lottery.co.uk, Lotto Odds
Powerball 1 in 292,201,338: Multi-State Lottery Association, Powerball prizes and odds
EuroMillions 5 of 50 plus 2 Lucky Stars of 12: EuroMillions, Odds of Winning
That players really do avoid recently drawn numbers, measured on actual ticket sales: C. T. Clotfelter and P. J. Cook, "The ‘Gambler’s Fallacy’ in Lottery Play." Management Science 39(12), 1521–1525, 1993.
Why a short run is expected to look like a long one, which is the error underneath it: A. Tversky and D. Kahneman, "Belief in the law of small numbers." Psychological Bulletin 76(2), 105–110, 1971.

Problem solved in full

Six numbers from 49 and a Superzahl from ten in German Lotto 5 steps
German Lotto: six numbers from 49, plus a Superzahl from ten. Work out the odds, then the honest question — how big must the jackpot be before a ticket is worth buying?
1. Order does not matter for the six main numbers, so it is a combination: the ordered count divided by the 720 ways the same six could arrive.
  
  $(6 49) = \frac{49 !}{6 ! 43 !} = 13983816$
2. The Superzahl is drawn separately, so it multiplies rather than adds. Ten independent outcomes on top of nearly fourteen million.
  
  $\times 10 Superzahl ⟹ 139838160 outcomes$
3. The expected return is what the operator pays back, and at 28.6% of a €1.20 ticket that is 34 cents. The rest is the loss.
  
  $E [return] = 0.286 \times 1.20 = 0.343 ⟹ E [profit] = - 0.857 per ticket$
4. So a ticket costs €0.857 in expectation, every time, and no strategy touches that number — picking birthdays, avoiding last week's numbers and buying in syndicates all leave it exactly where it is.
5. For the expectation to break even, the jackpot alone would have to make up the shortfall, which needs roughly €135 million.
  
  $break-even : J \times \frac{1}{139 838 160} must cover the shortfall \Rightarrow J \approx 134.9 M$
Answer

The tool prints 1 in 139,838,160, a 28.6% return rate and an expected loss of €0.857 per ticket. Two consequences are worth more than the odds. First, the break-even jackpot of about €134.9M is sometimes reached — and even then the ticket is not a good bet, because a shared jackpot halves the prize while the odds stay put. Second, one ticket a week gives a 50% chance of the jackpot after about 1.9 million years. The number that sells tickets is the prize; the number that decides them is the 28.6%, and it is printed on no ticket anywhere.

$N = (6 49) \times 10 = 139838160; E = - 0.857 per 1.20 ticket$

By the 42Math team · Updated 2026-05-14

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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