Expected value, in full
Written out, the expected value of X is the sum of x times P(x) taken over every outcome. The probabilities have to cover the whole sample space and add to 1. The answer is an average across the long run, not a forecast for any single trial. A fair d6 has expected value (1+2+3+4+5+6)/6 = 3.5, a value the die can never actually show, which is a useful reminder that an expectation is a summary and not a prediction.
For a lottery ticket the natural quantity is money. Expected value per ticket is the sum over prize tiers of prize times the probability of hitting that tier, minus the price of the ticket. Games are built so this comes out negative. Draw games commonly return around half of stakes as prizes, with the rest going to good causes or state funds, retailers, tax and the operator, so the expected loss per ticket runs close to half its price.
Rollovers complicate the picture without rescuing it. Once a jackpot grows large enough, the nominal expected value of a ticket can cross zero. Three things usually pull it back under. Advertised jackpots are often annuity totals, and the cash option is a substantially smaller lump sum. Winnings may be taxed. And a large jackpot sells more tickets, which raises the chance of a shared win and cuts the jackpot term of the sum proportionally.
- Probability
- EV, Expectation, Mean
A worked example
A small game: pick 3 numbers from 10, ticket price $1. There are C(10,3) = 120 equally likely draws. Matching all three pays $50 at probability 1/120. Matching exactly two pays $2, and there are C(3,2) x C(7,1) = 21 such tickets, probability 21/120. Expected return is 50/120 + 42/120 = 92/120, about $0.767. Subtract the $1 stake and the expected value is about minus 23 cents per ticket.
Sources
Around 53% of UK National Lottery ticket revenue goes to the prize fund, with the rest split between good causes, lottery duty, retailers and the operator.
Wikipedia — National Lottery (United Kingdom)
See also
Where it comes up on this site
Expected value: frequently asked
Nominally, yes. In a large enough rollover the jackpot term can outweigh the ticket price on paper, which is the only case where the question is worth asking at all. Three things usually drag it back under. The advertised jackpot is an annuity total, so the cash a winner actually collects is smaller. Winnings are taxed. And a big jackpot sells more tickets, which raises the chance of splitting it. Even where the sum stays positive, the individual ticket remains an almost certain loss.
No. It is an average over an enormous number of plays, and lottery payouts are skewed enough that nearly every ticket lands on nothing. It describes the game, not the slip in your hand.
Multiply each prize by the probability of winning it, add the products, subtract the ticket price. Published odds tables supply the probabilities. Use the cash value rather than the advertised annuity figure if the lump sum is what you would take.