Editorial & Research Standards
This article was researched and written by the Lotto Pick HQ Editorial Team using verified public records from the Multi-State Lottery Association (MUSL) and state open data portals. All calculations reflect independent probability theory and are reviewed against our methodology and responsible play policy.
Understanding how lottery odds are calculated requires understanding hypergeometric distributions and combinatorics. When a drawing machine selects 5 white balls without replacement and 1 special ball, every single tier's odds can be exactly derived.
Combinatorics 101: The Binomial Coefficient
C(n, k) = n! / (k! × (n - k)!)
For Powerball (choosing 5 balls from 69): C(69, 5) = (69 × 68 × 67 × 66 × 65) / (5 × 4 × 3 × 2 × 1) = 11,238,513. Multiplying by the 26 possible Powerballs yields:
Total Powerball Combinations = 11,238,513 × 26 = 292,201,338
Complete 9-Tier Powerball Odds Table
| Match Tier | Base Prize | Exact Combinatoric Formula | Exact Odds |
|---|---|---|---|
| 5 + PB (Jackpot) | Jackpot | C(5,5) × C(64,0) × 1 | 1 in 292,201,338 |
| 5 + 0 | $1,000,000 | C(5,5) × C(64,0) × 25 | 1 in 11,688,054 |
| 4 + PB | $50,000 | C(5,4) × C(64,1) × 1 | 1 in 913,129 |
| 4 + 0 | $100 | C(5,4) × C(64,1) × 25 | 1 in 36,525 |
| 3 + PB | $100 | C(5,3) × C(64,2) × 1 | 1 in 14,494 |
| 3 + 0 | $7 | C(5,3) × C(64,2) × 25 | 1 in 579.8 |
| 2 + PB | $7 | C(5,2) × C(64,3) × 1 | 1 in 701.3 |
| 1 + PB | $4 | C(5,1) × C(64,4) × 1 | 1 in 91.98 |
| 0 + PB | $4 | C(5,0) × C(64,5) × 1 | 1 in 38.32 |
The overall odds of winning any prize in Powerball are 1 in 24.87.
Modern Mega Millions Odds
Under the modern 5/70 + 1/24 matrix, total combinations are C(70, 5) × 24 = 12,103,014 × 24 = 290,472,336. Overall odds of winning any prize are 1 in 22.8.