The Math Behind Lotteries: Expected Value Explained
The odds of winning the Powerball jackpot are 1 in 292,201,338. If you bought 104 tickets a year (two every draw) for 50 years, your chance of ever hitting the jackpot would be about 0.0018%. That's roughly 1 in 56,000.
Yet when the jackpot passes a billion dollars, people point out that the ticket's "expected value" is now positive. Sometimes that's even true on paper. It's almost never true in practice. Let's see why.
The Numbers You Pick Don't Matter, Who Else Picks Them Does
Every combination of numbers has exactly the same probability of being drawn. 1-2-3-4-5 is exactly as likely as any "random-looking" set.
What does change is how many other people picked the same numbers. Birthdays cluster below 31, and patterns on the play slip are popular. If you win with commonly chosen numbers, you're far more likely to split the jackpot. You can't change your odds of winning, but you can change your expected payout if you do win.
Where 1 in 292,201,338 Comes From
Powerball draws 5 white balls from 69 and 1 red Powerball from 26. Order doesn't matter for the white balls, so we count combinations:
C(69, 5) = 69! / (5! × 64!) = 11,238,513
Multiply by the 26 possible Powerballs:
11,238,513 × 26 = 292,201,338
That's the whole calculation. The same method gives the classic 6-from-49 lottery used in many countries its odds of 1 in 13,983,816. The factorials behind these numbers grow fast. For perspective, 69! has 99 digits.
Expected Value: What Is a Ticket Worth?
Expected value (EV) is the probability-weighted average of all outcomes:
EV = Σ P(outcome) × Value(outcome)
Powerball has nine prize tiers. Ignoring the jackpot, the eight smaller prizes (from $4 up to $1 million) add up to an expected value of about $0.32 per ticket. With the overall chance of winning any prize at about 1 in 24.87, most of that value comes from the $4 and $7 prizes.
A ticket costs $2. So for the ticket to break even, the jackpot has to contribute about $1.68:
Jackpot × (1 / 292,201,338) ≥ $1.68
Jackpot ≥ ≈ $491 million
So an advertised jackpot above about $491 million looks like a positive-EV bet. But that's before reality steps in.
Three Things That Shrink the Jackpot
1. The Advertised Number Is an Annuity
The headline jackpot is the total of 30 annual payments that grow over time. The cash option, a lump sum paid now, is typically around 45%–50% of the advertised figure. Right away, a "$1 billion" jackpot is worth under $500 million in present-day cash.
2. Taxes
In the U.S., federal withholding on large lottery wins is 24%, and the top federal rate is 37%. Many states tax winnings too. A lump sum can lose around 40% to taxes.
3. Splitting
Big jackpots attract many more ticket buyers. More tickets mean a higher chance that someone else matches the same numbers. If N other tickets are in play, the expected number of co-winners is roughly N / 292,201,338. With 300 million tickets sold, you'd expect about one other winner.
Put these together and a $1.5 billion advertised jackpot can shrink to an expected after-tax, after-split value well under $400 million, back below break-even.
An Insider Reference: The Man Who Bought Every Combination
Romanian-born economist Stefan Mandel noticed that when a lottery's jackpot exceeded the cost of buying every combination, the EV turned positive. He reportedly won 14 lotteries using this approach.
His most famous play came in February 1992, when his investor group targeted the Virginia Lottery. Its 6-from-44 format had 7,059,052 combinations, at $1 each, and the jackpot was about $27 million. The group managed to buy roughly 5 million tickets before time ran out and still held the winning ticket. Lotteries responded by banning bulk purchases and switching to formats with far more combinations, like Powerball's 292 million.
Two Concepts Worth Knowing
Combinations vs. Permutations
A permutation counts arrangements where order matters. A combination ignores order. Because lottery draws don't care about order, combinations are the right tool, and there are 5! = 120 times fewer of them than permutations of 5 balls.
Variance and Utility
Even at positive EV, a lottery ticket has enormous variance: almost certainly $0, with a tiny chance of a fortune. Economists use expected utility instead of expected value to explain why people buy tickets anyway: the small cost barely registers, while the dream has real value to the buyer.
Quick Answer: What Is the Expected Value of a Lottery Ticket?
The expected value of a lottery ticket is the sum of each prize multiplied by its probability, minus the ticket price. For a $2 Powerball ticket, non-jackpot prizes are worth about $0.32. After the cash option, taxes and split jackpots, the EV is negative in nearly every drawing.
Try Them Yourself
- Random Number Generator: draw your own "lottery" numbers
- Scientific Calculator: calculate C(69, 5) with factorials
- Statistics Formulas: expected value and probability rules
- Algebra Formulas: combinations and permutations
- Number to Words: hear how large 292,201,338 really is
- The Mathematics of Gambling: house edge and variance
Next time the jackpot makes headlines, compute its EV after the cash option and a 40% tax. It's a great exercise in how quickly a headline number shrinks.