Lottery Odds Calculator

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

Two different break-even points get quoted, and mixing them up is where most “the jackpot is finally worth it” arguments go wrong:

• Advertised, ignoring tax and the lump-sum discount: $2 × 292,201,338 = $584 million. This is the number the calculator shows, and it is the one you will see repeated online.
• What you would actually bank: the lump sum is roughly 60% of the advertised annuity, and federal tax takes 37% of that. Working backwards, the advertised jackpot has to clear about $1.7 billion before the cash in your account breaks even on a $2 ticket.

Neither figure accounts for the fact that huge jackpots draw huge ticket sales, so the chance of splitting the prize rises exactly when the prize is worth chasing.

Secondary prizes: Matching all 5 white balls but missing the Powerball wins $1,000,000 at odds of 1 in 11,688,054. (Not 1 in 11.24 million, which is the count of white-ball combinations. You also have to miss the Powerball, and that extra condition makes the outcome slightly rarer.) Matching 4 of 5 plus the Powerball pays $50,000 at 1 in 913,129.