Expected Value: The Number That Predicts Luck

How to price a gamble — from raffle tickets to the decisions that quietly matter more

Last spring I stood in a school gym holding two raffle tickets and did the math that killed the fun. The prize was a $300 gift card, 400 tickets were sold, and each ticket cost $2. My expected payout was 300 × (1/400) = $0.75 — a ticket worth half its price. I bought two anyway, we lost, and my daughter asked the sharpest question of the night: why did you pay for a losing bet? The honest answer is that nobody had ever taught me to price luck. That missing skill has a name: expected value (EV), one small formula that turns “I feel lucky” into a number you can argue with. This guide is about the part everyone skips: “expected” is a forecast, not a promise — and it is not the same as an average. The counting and probability rules guides taught us how likely an outcome is; EV is what you do with that likelihood once you know it.

What Expected Value Actually Is

Expected value is the same recipe in every language: for every possible outcome, multiply its probability by its value, then add everything up.

EV = (prob₁ × value₁) + (prob₂ × value₂) + ... for every outcome
A die paying $6 only on a roll of 6:
EV = (1/6 × $6) + (5/6 × $0) = $1

Roll that die 600 times and the average payment settles near $600 — one dollar a roll. Nobody knows the next roll; everyone knows the next six hundred. That is the entire trick: EV takes a situation where any single outcome looks like pure luck and finds the slow, steady pull underneath it. The word “weighted” matters more than the word “careful” — each outcome does not vote equally. The $6 result gets one-sixth of the vote; the zeros get five-sixths.

The Fair Price of a Game

Turn that die into a game: it costs $1 to roll, you win $6 on a 6. The EV of what you are handed is $1, so a $1 entry fee is a fair price — on average the player wins nothing and loses nothing. Charge $2 and the player slowly drains their pocket; charge 75 cents and the game quietly gives money away. The moment I understood this, every carnival booth in town rearranged itself: the prizes were never the story — the price was the stream of probabilities running underneath.

A positive EV is a present someone is handing you. A negative EV is a rent you pay to be entertained. A fair EV is the polite center. Almost nothing in real life offers positive EV for free, because the people who operate games, insurance, and raffles price them so the number leans their way — that lean is the house edge.

A bet on red in European roulette: 18/37 × $1 − 19/37 × $1 = −$0.027
A typical $5 scratch ticket: EV ≈ $2.50 in winnings
A fair die game above: EV = $0 profit (fair by design)

Where the House Edge Hides in Everyday Life

Nobody gambles more carefully than the businesses we trust. An insurer sets a premium a tick above the expected claim, and the policy becomes profitable while remaining individually affordable. A phone store pushes a $3-per-month screen plan with an expected payout well under that. A school raffle sells $2 tickets for a $300 prize with 400 tickets — the institution wins the EV, families win the night out. None of this is evil; it is arithmetic with a price tag.

My worst personal bet was an extended warranty on a tablet. My repair estimate was $180; the four-year plan cost $60; I figured roughly a 1-in-10 chance I would ever claim. Expected claim value: 0.1 × $180 = $18 against the $60 premium. The math said skip it. I bought it anyway, out of fear, and we never did claim. EV did not make the decision for me — it simply split the decision into the part that was fear and the part that was number. That split, not any single answer, is the real payoff.

Slip Points That Predict Wrong Answers

Confusing EV with the most likely outcome. On the $6 die, the most likely single result is $0; the EV is $1. One is a forecast, the other a single cast of luck. Answering “what will I actually get?” with the EV is the classic error.
Forgetting an outcome. The probabilities must cover every branch and add to exactly 1. Skip the “nothing happens” branch — it often carries the biggest probability and quietly capsizes the total — and EV lies upward.
Treating EV as a plain average. A mean weights every value the same; EV weights each value by its probability. For equally likely outcomes they coincide, which is why the habit is tempting — and why it breaks exactly when outcomes are not equally likely.
Reading EV as a promise. Your one raffle ticket still loses 399 times out of 400. EV only earns its keep across many repetitions — one die roll is luck, a thousand are a salary.

The first mistake is the one I see on family game nights constantly: someone wins a big roll, high-fives the room, and declares the game “good.” One win is a fact; the EV of the GAME is the number that actually mattered all along.

Quick Practice Problems

A coin flip costs $1: heads wins $4, tails wins $0. What is the EV of your profit each play?
Answer: winnings EV = 0.5 × $4 = $2; minus the $1 fee = +$1 per play. Take this game.
100 tickets at $1 each, one prize of $75. What is the EV of a single ticket’s winnings?
Answer: (1/100) × $75 + (99/100) × $0 = $0.75. A ticket profits about −$0.25 on average.
Roll two dice: win $5 if the sum is 7, $0 otherwise, $1 to play. EV of profit?
Answer: p(7) = 6/36 = 1/6; winnings EV = (1/6) × $5 ≈ $0.83; profit ≈ −$0.17. A small house edge — skip it.
A $2 raffle with 400 tickets prizes $300. Fair price check?
Answer: EV of winnings = 300/400 = $0.75, so a fair ticket is $0.75 — you are paying $1.25 of fun.

Frequently Asked Questions

If EV is negative, should I never play?

Not automatically. Entertainment, a cause you believe in, or a night at the fair all carry value the calculator cannot see. EV tells you the price of the fun, not whether the fun is worth it. The trap is paying a negative EV while telling yourself the odds are even.

Is expected value the same as an average?

It is a weighted average. The plain mean gives every value an equal vote; EV gives each outcome a vote equal to its probability. They agree only when all outcomes are equally likely — which is why the shortcut feels right until the moment it is wrong.

Why do casinos always seem to win?

The law of large numbers. A negative EV on every single bet averages out perfectly across tens of thousands of plays. The casino does not need to win your roll — it only needs many rolls, and the number it set quietly does the rest.

Does EV show up in school math?

It is the natural bridge between the probability rules and algebra: list the outcomes, assign chances, build the weighted sum. Every “is this offer any good?” story — insurance, sales, even gacha games — is the same skeleton.

Price one real gamble this week. The next raffle ticket, the next “double or nothing,” the next warranty — write the two columns (outcome × probability), total them, and let the number argue with your gut before your money joins the debate. Practice your own EV problems in our free math tutor apps or bring a tricky “is this actually a good bet?” question to Math Q&A and we will price it together.