Lottery Odds Calculator
Calculate the exact odds of winning any lottery.
Enter the numbers to pick, total pool, and optional bonus ball to see your probability and expected value.
The lottery is a tax on mathematical ignorance, but understanding the actual odds and expected value (EV) makes the entertainment cost transparent. Lotteries use combinations (not permutations) since the order of numbers drawn doesn’t matter.
Odds of winning (combinations formula):
C(n, k) = n! / (k! × (n − k)!)
For a lottery where you pick k numbers from n possible numbers:
Odds of jackpot = 1 / C(n, k)
Expected Value formula:
EV = (Jackpot × Probability of Winning) − Ticket Price
What each variable means:
- n — total numbers in the pool (e.g., Powerball picks 5 from 69)
- k — numbers you must match (e.g., 5 main numbers + 1 Powerball from 26)
- C(n, k) — total number of ways to choose k items from n (number of possible combinations)
- Jackpot — the advertised prize, but lump sum is typically 60% of advertised amount; after 37% federal tax: ~38% of advertised figure
- EV — positive EV means expected profit; virtually all lotteries have negative EV by design
Worked example — Powerball: Main game: pick 5 from 69 → C(69,5) = 11,238,513 combinations Powerball: pick 1 from 26 → 26 combinations Total jackpot combinations = 11,238,513 × 26 = 292,201,338
Odds = 1 in 292,201,338 (about 1 in 292 million)
Jackpot advertised: $500 million Lump sum (~60%): $300 million After 37% federal tax: $189 million EV = ($189,000,000 / 292,201,338) − $2 = $0.647 − $2.00 = −$1.35 per $2 ticket
Even at $500M advertised, the EV is deeply negative. The lottery only becomes marginally positive EV (before tax) when jackpots exceed ~$900M — but that ignores shared jackpots when many people play, which further reduces EV.
Secondary prizes: Matching 5 of 5 (no Powerball) wins $1,000,000 at odds of 1 in 11.2 million. Matching 4 of 5 + Powerball: $50,000 at 1 in 913,129.