Expected Value Calculator

Should you buy the raffle ticket? Is the insurance premium fair? Is this game rigged? Expected value is the single number that answers questions like these honestly. It weighs every possible outcome of a random event by how likely that outcome is, producing the average result you would see if you repeated the event many, many times. The expected value calculator does this weighting for you: supply the outcomes and their probabilities, and it returns the long-run average — the figure that separates a genuinely good bet from an entertaining way to lose money slowly.

Quick Answer

Expected value is EV = x₁p₁ + x₂p₂ + … + xₙpₙ — multiply each outcome by its probability and add the products. The probabilities must add up to exactly 1. If playing costs money, subtract the cost from the payout EV to get the net expected value.

Expected Value, in Plain Terms

Think of a fair die. Its faces are worth 1 through 6, each with probability 1/6, so the expected value of one roll is (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 3.5. You will never roll a 3.5, yet if you rolled the die a thousand times and averaged the results, your average would land very close to 3.5. That is the whole idea: expected value is not a prediction of the next trial, it is the centre of mass of all trials, the value the law of large numbers drags your running average toward.

The probabilities do the steering. An outcome worth a fortune counts for almost nothing if its probability is one in a million, while a modest outcome with probability one half dominates the calculation. This is why expected value is the natural language for comparing gambles, pricing insurance, valuing a lawsuit settlement, or deciding whether a business risk is worth taking: it forces every outcome to be discounted by how real it is.

Insurance shows both sides of the ledger at once. An insurer covering a house against a 1-in-500 annual chance of a $200,000 loss faces an expected claim cost of 200,000 × (1/500) = $400 a year on that policy, and must charge more than $400 to cover expenses and profit. The homeowner who pays, say, $700 has a negative expected value in raw dollars — yet the purchase can still be rational, because the uninsured outcome is ruinous rather than merely expensive. Expected value measures the average, not the wisdom, and it is at its most powerful combined with an honest look at how bad the worst outcomes are.

How to Use the Expected Value Calculator

1. List every possible outcome of the event — including the dull ones where nothing happens or you simply lose your stake.
2. Assign each outcome a value: money won, points scored, or whatever quantity you are averaging, using negative values for losses.
3. Assign each outcome its probability as a decimal or fraction.
4. Check that the probabilities add up to 1 before calculating; if they do not, an outcome is missing or a probability is wrong.
5. Run the calculation and read the expected value.
6. If there is an entry cost, subtract it from the result to get the net expected value per play — the number that decides whether the game favours you.

Worked Example: A Carnival Dice Game

A stall charges $3 to roll one fair die. Rolling a 6 pays out $12, rolling a 5 pays out $4, and any other face pays nothing. Is the game fair?

First work out the expected payout. A 6 comes up with probability 1/6 and pays 12, contributing 12 × (1/6) = $2.00. A 5 also has probability 1/6 and pays 4, contributing 4 × (1/6) ≈ $0.67. The remaining four faces, with combined probability 4/6, pay 0 and contribute nothing. Total expected payout: 2.00 + 0.67 = $2.67 per roll.

Now subtract the $3 ticket: 2.67 − 3.00 = −$0.33 per game. The game is tilted against the player by about 33 cents a roll. One game proves nothing — you might roll a 6 immediately — but over 300 games the expected loss is about 300 × 0.33 ≈ $100, and the stall owner can bank on collecting it.

• Coin-flip game: win $5 on heads, lose $3 on tails → EV = 0.5(5) + 0.5(−3) = +$1.00 per flip
• Average of one fair die roll → 3.5
• Raffle: 1-in-1,000 chance of a $900 prize, ticket $1 → payout EV = $0.90, net EV = −$0.10
• A certain outcome: probability 1 of receiving $50 → EV = $50, as expected

How to Read a Positive, Zero, or Negative Result

A positive net EV means the game or decision favours you in the long run; negative means it favours the other side; zero means it is fair — neither side has an edge. Three cautions keep the number honest. First, probabilities that fail to sum to 1 are the most common input error and poison everything downstream. Second, expected value says nothing about risk: a coin flip between winning $1,000 and losing $1,000 has the same EV of zero as receiving nothing at all, but the two feel very different, and the spread of outcomes is a separate question. Third, a single trial — or ten — can land far from the EV; the average only asserts itself over many repetitions, which is why casinos love it and one-off players should respect it. Lotteries lean on this hardest of all: a jackpot outcome can be worth millions on paper while contributing only cents to the ticket's expected value, because its probability is measured in hundred-millions.

Pair Expectation With Spread

Expected value tells you the centre; you often need the probabilities first and the spread second. The Probability Calculator helps you work out the chances of the outcomes you are weighing, and the Standard Deviation Calculator measures how wildly results swing around the average. Together with the rest of the math tools, they turn a gut feeling about a gamble into arithmetic you can argue with.