Lottery Odds, Explained
After reading this you can compute the exact odds of every match tier in a pick-k-of-N lottery, and decide whether a jackpot is large enough to make a ticket worth the price on jackpot alone.
What lottery odds actually measure
A standard lottery draws k distinct numbers from a pool of N. You buy a ticket with your own k numbers. The draw is a random selection with no order and no replacement, so every possible combination of k numbers is equally likely. The number of those combinations is what sets your odds.
Take the classic 6-of-49 game. There are exactly 13,983,816 ways to choose 6 numbers from 49. Only one of them matches all six drawn numbers. So the jackpot odds are 1 in 13,983,816, or about 0.00000715%. That single number is the hook: if you buy one ticket a week, you would expect to hit the jackpot roughly once every 268,919 years.
Most lotteries also pay smaller prizes for matching some but not all of the numbers. This calculator lists every tier, from matching all k down to matching the smallest paying count, with the exact odds for each.
The counting formula
The total number of tickets is the binomial coefficient, written \binom{N}{k} and read "N choose k".
Here N! is the factorial of N (the product of all integers from 1 up to N), and the same for k! and (N-k)!. The formula counts every unordered group of k numbers exactly once. For 6-of-49 it gives \binom{49}{6} = 13{,}983{,}816.
To count tickets that match exactly m of the drawn numbers, split the pool into two groups: the k numbers that were drawn, and the N-k that were not. You need m from the drawn group and the remaining k-m from the not-drawn group.
The probability of matching exactly m is that count divided by the total.
This is the hypergeometric distribution. The three binomial terms are: \binom{k}{m} choosing your hits from the drawn numbers, \binom{N-k}{k-m} choosing your misses from the rest, and \binom{N}{k} the total tickets.
Many real lotteries add a bonus ball or a separate second pool (Powerball, EuroMillions). Those change the tier structure and the odds. The formula here covers the pure k-of-N single-pool game. Treat its numbers as the base case, then adjust for extra balls using the game's own rules.
Worked example: 6-of-49 with a 5 million jackpot
Reproducing the demo
Use N = 49, k = 6, jackpot 5,000,000, ticket price 2. These are the demo button values.
- Total tickets: \binom{49}{6} = 13{,}983{,}816.
- Match 6: \binom{6}{6}\binom{43}{0} = 1. Odds 1 in 13,983,816.
- Match 5: \binom{6}{5}\binom{43}{1} = 6 \times 43 = 258. Odds 1 in 54,201.
- Match 4: \binom{6}{4}\binom{43}{2} = 15 \times 903 = 13{,}545. Odds 1 in 1,032.
- Match 3: \binom{6}{3}\binom{43}{3} = 20 \times 12{,}341 = 246{,}820. Odds 1 in 56.66.
The jackpot-only expected value takes only the top tier into account. The chance of the jackpot is 1 / 13,983,816. Multiply by the 5,000,000 prize:
So the jackpot alone returns about 0.36 per ticket against a 2.00 price. On jackpot value only, you lose about 1.64 per ticket. The smaller tiers add some return, but at a 5 million jackpot they do not close that gap.
How large must the jackpot be
The jackpot-only EV equals the price when the jackpot equals the total number of tickets times the price. For 6-of-49 at 2 per ticket that break-even jackpot is
Below about 28 million, the jackpot term alone cannot repay the ticket. This is a ceiling on how good the bet can look, not a promise. Real jackpots are split among multiple winners, taxed, and often paid as an annuity worth less than the headline figure. Each of those pushes the true break-even higher.
Reading and interpreting the results
The odds are best read as "1 in X". Match 3 in 6-of-49 happens about 1 in 57 tickets, which is common enough that you will see it. Match 5 at 1 in 54,201 is rare enough that a weekly player might wait a thousand years. Do not confuse a low-tier prize being likely over many draws with any single ticket being likely to win.
A useful check: the probabilities of all possible match counts (including the tiers that pay nothing) sum to exactly 1. For 6-of-49 the chance of matching 2 or fewer is about 98.14%, which is why most tickets win nothing.
| Match | Winning combinations | Odds (1 in) | Probability |
|---|---|---|---|
| 6 | 1 | 13,983,816 | 0.0000000715 |
| 5 | 258 | 54,201 | 0.00001845 |
| 4 | 13,545 | 1,032 | 0.0009686 |
| 3 | 246,820 | 56.66 | 0.01765 |
Common mistakes
Buying two different tickets doubles your jackpot chance to 2 in 13,983,816, not to anything meaningful. Ten tickets give 10 in 13,983,816, still about 1 in 1.4 million. Coverage grows linearly and starts from almost nothing.
Three errors show up constantly. First, treating past draws as informative: each draw is independent, so a number that has not appeared in 40 draws is no more likely next time. Second, thinking a "lucky" combination like 1-2-3-4-5-6 is less likely than a scattered set. Every specific combination has the same 1 in 13,983,816 chance. The scattered set only feels more likely because there are more scattered patterns overall.
Third, ignoring jackpot splitting. Popular number patterns get chosen by many players, so if such a combination wins, you share the prize. That lowers your effective EV even when the headline jackpot is high.
When this calculator helps and when it does not
Use it to get exact tier odds for any single-pool k-of-N game, to compare two game formats, and to find the jackpot level where a ticket stops being a guaranteed loss on jackpot value alone. It is honest arithmetic, not a strategy for winning.
Do not use the jackpot-only EV as a full expected value. It leaves out every lower prize, and it also leaves out annuity discounting, taxes, and prize splitting. The real EV of a ticket is almost always well below the price. If you want to reason about a betting decision where you actually have an edge, the odds-versus-price logic in Poker Pot Odds and Outs is a closer fit, because there the math can favor you.
Frequently asked questions
Why are the odds 1 in 13,983,816 for 6-of-49?
Because there are exactly that many ways to pick 6 numbers from 49, and only one of them is the winning set. The count comes from \binom{49}{6}.
Do quick picks and chosen numbers have different odds?
No. A random quick pick and hand-chosen numbers are both single combinations, each with identical odds. Choosing numbers only affects how likely you are to share a prize with other players.
Does a bigger jackpot improve my odds?
No. The odds depend only on N and k. A larger jackpot raises the payoff if you win, which raises expected value, but the probability of any tier is unchanged.
What is the smallest jackpot worth a 2 ticket in 6-of-49?
On jackpot value alone, about 28 million, since 13,983,816 x 2 = 27,967,632. Splitting, taxes, and annuities push the honest figure higher.
Why do most tickets win nothing?
In 6-of-49, matching 2 or fewer numbers pays nothing and happens about 98.14% of the time. The paying tiers together cover under 2% of tickets.
Related tools
Blackjack Strategy & EV Explorer
Exact basic strategy charts and EVs for your exact rules.
Card Counting Trainer
Learn Hi-Lo card counting: speed drills, true-count quizzes and a bankroll simulator with your bet spread.
Roulette Odds
Win probability, payout and house edge for every bet.
Blackjack House Edge
House edge from the exact table rules you are playing.
Craps Odds
Exact win probability and house edge for the main bets.
Expected Loss Calculator
The honest cost of a session at any casino game.