Why winning Powerball is 1 in 292 million and how expected value proves the house wins.
Combinations vs Permutations
A permutation counts arrangements where order matters; a combination counts selections where order does not matter.
Combination Formula: nCr = n! / [ r! (n - r)! ]
Calculating Total Powerball Combinations
Choose 5 white balls from 69: 69C5 = 11,238,513. Multiply by 26 red Powerball numbers: 11,238,513 × 26 = 292,201,338.
Jackpot Odds: 1 in 292,201,338 (0.000000342%)
Mathematical Expected Value (EV)
Expected value multiplies the probability of each prize by its payout, minus the ticket cost.
EV Formula: EV = Σ [ P(Prize_i) × Payout_i ] - Ticket_Cost
Why the House Always Wins
Even with massive billion-dollar jackpots, income tax withholdings (37%+) and the risk of splitting the jackpot with multiple winners keep long-term EV negative.
Mathematical Reality: Lotteries function as a voluntary probability tax.
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