The Expected Value of a Lottery Ticket
A clean derivation: when is the expected return of a Powerball ticket positive? (Almost never.)
Expected value (EV) is the probability-weighted average of all outcomes. It answers: "If I played this many times, what would I gain or lose per play, on average?"
The formula
For Powerball, the prize tiers and their probabilities are public. The jackpot dominates when the jackpot is large; smaller fixed prizes (e.g., $1M for matching 5, $4 for matching the Powerball alone) contribute a roughly constant amount.
A worked example
Suppose the advertised jackpot is $300 million. The jackpot probability is 1 / 292,201,338. The expected contribution from the jackpot tier:
≈ $1.027 per $2 ticket
Adding contributions from the eight smaller prize tiers (roughly $0.32 per ticket, dominated by the $4 match-Powerball prize), total expected prize value is about $1.35. Ticket cost is $2:
That's the gross EV. After taxes and the lump-sum vs. annuity adjustment, real EV is sharply more negative.
The lump-sum and tax adjustments
Advertised jackpots are 30-year annuities. The lump-sum cash option is typically 50–60% of that. Federal tax: 37% top bracket. State tax: 5–10% in most states. So the after-tax cash value of a "$300M jackpot" is roughly:
≈ $96 million net
Re-running the EV with $96M as the prize:
EV ≈ $0.33 + $0.32 − $2.00 = −$1.35
A more realistic EV per $2 Powerball ticket, after taxes and lump-sum, is about −$1.35 to −$1.65. You lose, on average, about 68–82 cents on every dollar spent.
When does EV go positive?
For a single ticket, EV becomes positive only when the after-tax lump-sum jackpot exceeds approximately:
≈ $584 million (gross / annuity)
But this ignores shared-jackpot risk. When the jackpot is very high, more people play; the probability you split your winnings rises. Realistic models suggest the shared-jackpot effect cancels EV-positivity even at $1 billion+ jackpots.