Expected Value of a Lottery Ticket: How to Calculate It, With Examples

By KT Apps · Updated October 16, 2026

About this guide

This is a maths and probability explainer. It does not recommend buying lottery tickets or any other form of gambling. All prize amounts below are invented for the examples. Lottery Simulator is a simulation for learning: it does not sell tickets or involve real money. Help resources are listed at the end.

What expected value means

For each possible outcome, multiply what you would win by the probability of that outcome, then add the results together. Subtract what you pay to play, and you have the expected value (EV) of one play:

EV = (prize 1 × probability 1) + (prize 2 × probability 2) + … − cost

Expected value is not what you will get on any single ticket. You will almost always get either nothing or a small prize. It is the long-run average per play, and the more times a game is played, the closer the average result tends to get to it.

Example 1: a simple raffle

Imagine a made-up raffle: 1,000 tickets are sold at $2 each, and there is one prize of $1,000.

1. List the outcomes: win $1,000, or win nothing.
2. Probability of winning with one ticket: 1 in 1,000, or 0.001.
3. Expected winnings: $1,000 × 0.001 = $1.00.
4. Subtract the cost: $1.00 − $2.00 = −$1.00.

On average, each ticket loses $1. That makes sense: the organiser takes in $2,000 and gives back $1,000, so half of every dollar staked is kept. Charity raffles are often set up this way on purpose, and many people buy tickets knowing that the money supports a cause.

Example 2: a lottery with several prize tiers

Now take Japan's Loto 7 format: 7 numbers from 37, with 7 main numbers and 2 bonus numbers drawn each time. The probability of each prize tier comes from counting combinations (see how lottery odds are calculated). Suppose, purely for illustration, a ticket costs $3 and the prizes are as follows:

• 1st prize, 7 main numbers, probability 1 in 10,295,472: invented prize $5,000,000
• 2nd prize, 6 main numbers plus a bonus number, about 1 in 735,391: invented prize $50,000
• 3rd prize, 6 main numbers, about 1 in 52,528: invented prize $5,000
• 4th prize, 5 main numbers, about 1 in 1,127: invented prize $50
• 5th prize, 4 main numbers, about 1 in 72: invented prize $10
• 6th prize, 3 main numbers plus a bonus number, about 1 in 42: invented prize $5
• Anything else: nothing

Multiply each prize by its exact probability (the number of winning tickets for that tier divided by 10,295,472):

• 1st: $5,000,000 × 1 ÷ 10,295,472 ≈ $0.49
• 2nd: $50,000 × 14 ÷ 10,295,472 ≈ $0.07
• 3rd: $5,000 × 196 ÷ 10,295,472 ≈ $0.10
• 4th: $50 × 9,135 ÷ 10,295,472 ≈ $0.04
• 5th: $10 × 142,100 ÷ 10,295,472 ≈ $0.14
• 6th: $5 × 242,550 ÷ 10,295,472 ≈ $0.12

Added without rounding, that is about $0.95 back for every $3 ticket, so EV ≈ $0.95 − $3.00 = −$2.05. Over many tickets, a player in this made-up game should expect to lose roughly two-thirds of what they spend. Notice that the small, frequent 5th and 6th prizes contribute about half as much as the jackpot does, even though they are tiny. Real lotteries have their own prize tables, but they are designed in the same spirit: part of the ticket money pays for running the game and, in many countries, goes to public causes, so the prizes paid out are less than the money taken in.

What about giant jackpots?

When a jackpot rolls over many times, people sometimes argue that a ticket has become a good bet. The simple calculation can look positive, but several things pull the real figure down:

• Shared prizes: big jackpots attract more players, which raises the chance that a winning ticket has to split the prize.
• Advertised amounts are often an annuity paid over many years; the lump-sum option is smaller.
• Taxes on winnings, depending on where you live.
• Variance: even with a favourable average, a single ticket almost certainly wins nothing. Expected value describes millions of plays, not one.

The practical lesson is that expected value tells you what the game costs on average, not whether you will be lucky.

Calculating versus simulating

Calculating

With a prize table and the probabilities, the calculation is exact and takes a few minutes with a calculator or spreadsheet. The catch is that you need accurate prize amounts, and many real prizes change each draw depending on sales and how many people win.

Simulating

A simulation buys imaginary tickets over and over and keeps a running balance. It gives an estimate rather than an exact figure, but it shows something the formula hides: the path. Balances wobble, jump after a lucky small win, and then drift downward. Seeing that drift over thousands of draws makes a negative expected value easy to grasp.

Seeing expected value in Lottery Simulator

Lottery Simulator uses the Loto 7 format with a ¥300 ticket and approximate example prizes for the six tiers: about ¥700,000,000, ¥6,077,700, ¥500,000, ¥6,500, ¥1,400 and ¥1,000. Using the tier probabilities above, those prizes give back about ¥134 for every ¥300 ticket, roughly 45%, so the expected value is about −¥166 per ticket. No money is involved at any point.

1. Tap Reset All to start with clean counters.
2. Tap Buy One a few times. Most draws win nothing: only about 1 ticket in 26 reaches any prize tier.
3. Tap Auto Buy and watch the Simulation Results panel: attempts, total purchase amount, total prize amount and balance update as the draws run.
4. Let it run for a while, then tap Stop. Compare total prizes with total purchases; the ratio is your simulated return per unit spent. Without a jackpot or 2nd prize it usually stays well below the 45% average, because those rare prizes make up more than half of it.
5. Try Hit First Prize to keep drawing until all 7 numbers match, and see how much imaginary money was spent along the way and how many years of weekly play that represents.

If you run the simulation several times, you will notice that the balance usually ends far below zero even in runs that include a jackpot. That is negative expected value in action.

Playing responsibly and getting help

If you choose to play real lotteries, treat tickets as paid entertainment with a fixed budget you can afford to lose, never as a way to make money or recover losses. If gambling is affecting your money, mood or relationships, free and confidential support is available: in the US, the National Council on Problem Gambling runs a national helpline; in the UK, GamCare runs the National Gambling Helpline. Elsewhere, search for your local gambling helpline.

Frequently asked questions