Lottery Math Explained
Deep dive into the mathematics behind lottery probability and odds calculation.
Mathematical Foundation
Combinations
Lottery odds use combinations (order doesn't matter) rather than permutations (order matters).
Probability
Understanding probability is essential for interpreting lottery odds.
Odds
Odds = Probability of losing / Probability of winning.
Combinations vs Permutations
The difference between combinations and permutations is crucial for understanding lottery odds. Because order doesn't matter in lottery draws, combinations are used.
Permutations (Order Matters)
If order mattered, selecting 3 numbers from 10 would give us:
P(10,3) = 10 × 9 × 8 = 720
Different arrangements count as different outcomes.
Combinations (Order Doesn't Matter)
Since order doesn't matter in lottery draws, combinations are used:
Same numbers in any order count as a single combination.
C(10,3) = 10! / (3! × (10-3)!) = (10 × 9 × 8) / (3 × 2 × 1) = 120
Expected Value
Expected value (EV) helps quantify the average outcome if you played the lottery many times:
EV = (P × J) - C P = probability, J = jackpot, C = cost per ticket.
Powerball Example
Let's calculate the odds for Powerball (5/69 + 1/26):
Main Numbers (5/69)
C(69,5) = 11,238,513
Possible combinations for main numbers.
Powerball (1/26)
C(26,1) = 26
Possible Powerball numbers.
Total Odds
1 in 292,201,338 chance of winning the Powerball jackpot.
Total Odds = 11,238,513 × 26 = 292,201,338 Multiple Tickets
Buying multiple tickets improves your odds linearly, but the expected value remains negative.
Improved Odds with Multiple Tickets
If you buy T tickets, your probability becomes:
Odds improve linearly with the number of tickets purchased.
P(win with T tickets) = T × P(single ticket) Probability with multiple tickets.