The number engineered so the casino always wins in the long run.
A $1 lottery ticket — what is this slip of paper really worth?
What remains on average — expected value.
Experiment
Hands-on experiment
Predict first — buy 10,000 $1 tickets (jackpot $1M at 1-in-8-million odds, etc.). How much comes back in winnings?
Buy 10,000 tickets (spend: $10,000)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Buy the ticket? Take the insurance? Make this investment? Life constantly demands comparisons between options whose outcomes are unknown.
Comparing requires a scale. Multiply each outcome's value by its probability and add — expected value summarizes an entire uncertain future in one number.
Casinos, insurers, and lotteries are all businesses engineered on this number. Someone who can compute EV and someone who can't are playing different games at the same table.
Insight
Insights from the video
“Expected value is the center of mass of a probability world.”
Picture the outcomes as weights on a number line, each weighing its probability. The point where the beam balances — that's the expected value.
“The house always wins — because the EV was engineered.”
Every casino game's EV is designed slightly negative. Anyone can win one round, but the law of large numbers guarantees the house's share over repetition.
Misconception
Common misconceptions
A negative expected value makes a choice automatically wrong.
EV is the average over long repetition. Insurance has negative EV yet can be rational because it removes the risk of ruin. In one-shot, high-stakes situations, variance (risk) matters too.
If EV is $0.50, you'll often receive around $0.50.
A ticket's EV of $0.14 can be an amount you literally never receive (mostly $0, very rarely a fortune). EV isn't a prophecy of the outcome — it's the balance point of repetition.
Formula
Writing it as math
Let's formalize the 'what remains on average' you verified over 10,000 tickets.
Definition of expected value
Each outcome's value times its probability, all summed. For a lottery: ($1M × 1/8M) + ($1,000 × 1/100k) + …
The link to the law of large numbers
As repetitions grow, the actual average converges to the EV. Why your 10,000-ticket runs always landed near it.
The fair price
The expected prize IS the fair price. The gap to the actual price is the seller's share (the house edge) — about 50% for lotteries.
In Real Life
Where you meet it in real life
Pricing insurance
Premium ≈ accident probability × payout + overhead. Negative EV for you, yet worth it for removing unbearable risk.
Casinos and lotteries
Roulette's expected return: −2.7%; lotteries: about −50%. Enjoying 'the price of fun' knowingly is different from not knowing.
Investment judgment
'+30% if it works (60%), −20% if not (40%)' — EV +10%. The basic frame of scenario-based investing.
Games and AI choices
A chess engine picks moves by each move's expected win rate. The standard tool for optimal choice under uncertainty.
Try Yourself
Test yourself
Q1Roll a die once and win the face value × $10. Is a $40 entry fee worth it?Show answer ▾
Expected prize = (1+2+3+4+5+6)/6 × $10 = $35. At $40 entry you lose $5 per round on average. At $35 or less the game turns favorable.
Q2A sure $900 vs a 10% shot at $10,000. Which has the higher EV? Which would YOU choose?Show answer ▾
EVs: $1,000 vs $900 — the gamble is higher. But it carries a 90% chance of nothing. If EV is the scale's reading, variance is the scale's wobble — a preview of the next concept.
Q3Is an insurer selling a product that loses money in expectation?Show answer ▾
The opposite — the EV is negative for the customer (premium > expected payout), and that gap is the insurer's share. Buying is still rational because of a value outside EV: eliminating ruin-level risk.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Bayes' Theorem
Having updated probabilities with evidence, it's time to decide with them.
← Bayes' Theorem labLeads to next
Variance & Standard Deviation
Two options with equal EV but different wobble are utterly different choices. The tool that measures the scale's wobble — variance — is the next station.
Go to the Variance & Standard Deviation lab →Related
Labs worth exploring together
Related lab
Probability
EV's raw material — first master counting each outcome's probability.
Go to the Probability lab →Related lab
Compound Interest
How expected returns grow over time — combine the two tools and you get investment judgment.
Go to the Compound Interest lab →