Lottery Probability Calculator
Calculate your odds of winning any lottery. Enter the number pool, balls drawn, and optional bonus ball to find jackpot probability, partial match odds, and expected value per ticket. See also our Combinations Calculator, Probability Calculator, and Expected Value Calculator.
How to Use the Lottery Probability Calculator
This calculator determines the exact mathematical odds of winning any lottery based on its format. Lotteries work by drawing k numbers from a pool of n without replacement. The number of possible combinations determines your odds. Some lotteries add bonus balls from a separate pool, which multiplies the total combinations and makes winning harder.
Enter the total numbers in the pool (e.g., 49 for a 6/49 lottery), how many numbers are drawn, and whether there's a bonus ball. For lotteries like Powerball (5/69 + 1/26), set total numbers to 69, numbers drawn to 5, bonus balls to 1, and bonus pool to 26. The calculator shows odds for the jackpot and all partial matches.
The expected value calculation shows the average return per ticket. For most lotteries, the expected value is negative (you lose money on average). A lottery becomes mathematically favorable only when the jackpot grows large enough to offset the astronomical odds — but even then, the probability of actually winning remains negligibly small for any individual ticket.
Formula
Jackpot Probability (no bonus):
P = 1 / C(n, k) = k!(n-k)! / n!
With Bonus Ball:
P = 1 / [C(n, k) × bonus_pool]
Partial Match (m out of k):
P(m matches) = C(k,m) × C(n-k, k-m) / C(n,k)
Expected Value:
EV = (jackpot × P_win) - ticket_cost
Tickets for 50% chance:
t = ln(0.5) / ln(1 - P_win)
Example Calculation
Lottery: Pick 6 numbers from 49 (like Canada Lotto 6/49)
Total combinations: C(49,6) = 49!/(6!×43!) = 13,983,816
Jackpot odds: 1 in 13,983,816
Probability: 0.00000715%
Match 5/6: C(6,5)×C(43,1)/C(49,6) = 258/13,983,816 = 0.00184%
Match 4/6: C(6,4)×C(43,2)/C(49,6) = 13,545/13,983,816 = 0.0969%
Match 3/6: C(6,3)×C(43,3)/C(49,6) = 246,820/13,983,816 = 1.765%
EV ($10M jackpot, $2 ticket): $10M/13.98M - $2 = -$1.28
Reference Table
| Lottery | Format | Jackpot Odds |
|---|---|---|
| Powerball (US) | 5/69 + 1/26 | 1 in 292,201,338 |
| Mega Millions (US) | 5/70 + 1/25 | 1 in 302,575,350 |
| EuroMillions | 5/50 + 2/12 | 1 in 139,838,160 |
| UK Lotto | 6/59 | 1 in 45,057,474 |
| Canada Lotto 6/49 | 6/49 | 1 in 13,983,816 |
| Australia Powerball | 7/35 + 1/20 | 1 in 134,490,400 |
| German Lotto | 6/49 + 1/10 | 1 in 139,838,160 |
| Japan Loto 6 | 6/43 | 1 in 6,096,454 |
Frequently Asked Questions
What are my actual chances of winning the lottery?
For major lotteries, your chances are astronomically small. Powerball odds are about 1 in 292 million — you're more likely to be struck by lightning (1 in 1.2 million), attacked by a shark (1 in 3.7 million), or become a movie star (1 in 1.5 million). Even buying 100 tickets per week for 50 years gives you only about a 0.09% lifetime chance of winning Powerball. The odds are designed to be nearly impossible for any individual.
Is the lottery ever a good investment?
Mathematically, almost never. The expected value of a lottery ticket is typically -40% to -60% of the ticket price (you get back 40-60 cents per dollar on average). Even when jackpots grow very large, the expected value rarely becomes positive because: (1) taxes take 30-50%, (2) lump sum is less than advertised, and (3) multiple winners split the prize. Lotteries are entertainment, not investment. The "investment" is the dream, not the return.
Does buying more tickets significantly improve my odds?
Buying more tickets increases your odds linearly, but from such a tiny base that it barely matters. Buying 10 Powerball tickets changes your odds from 1 in 292 million to 10 in 292 million (1 in 29.2 million) — still essentially zero. You'd need to buy about 202 million tickets ($404 million) to have a 50% chance of winning. The only way to guarantee a win is to buy every combination, which costs more than most jackpots are worth after taxes.
Why do some lotteries have worse odds than others?
Odds depend on the pool size and numbers drawn. Larger pools and bonus balls create more combinations. Powerball (5/69 + 1/26) has 292 million combinations because you need C(69,5) × 26 = 292,201,338. A simpler 6/49 lottery has only C(49,6) = 13.98 million combinations. Lottery operators design formats to balance jackpot size (harder odds = bigger jackpots from rollovers) against player engagement (some wins needed to maintain interest).
Are some numbers luckier than others?
No. In a fair lottery, every number has exactly the same probability of being drawn. However, choosing less popular numbers (above 31, since many people use birthdays) means you're less likely to share the jackpot if you win. Numbers like 7, 11, and 13 are commonly chosen, so winning with them means splitting with more people. The probability of winning doesn't change, but the expected prize does.
What is the difference between odds and probability?
Probability is the chance of winning expressed as a fraction: P = favorable/total (e.g., 1/13,983,816). Odds are expressed as a ratio of losing to winning: "1 in 13,983,816" means for every 1 win, there are 13,983,815 losses. In everyday language, "odds" and "probability" are used interchangeably, but technically odds = P/(1-P). For very small probabilities (like lotteries), the numerical difference is negligible.