Expected value is the average result over very many repetitions. Multiply each outcome by its probability and add.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A weighted average
Likely outcomes count more heavily than unlikely ones. That weighting is the whole idea.
You may never see it
A die has expected value 3.5, a face it does not have. Expected value describes the long-run average, not any single result.
Fair games
A game is fair when the expected gain is zero. Casino and lottery games are deliberately negative for the player.
Why negative games still get played
Insurance has negative expected value for the buyer, who accepts it to avoid a rare catastrophic loss.
Include the cost
When a game charges to play, subtract the cost. Forgetting it turns a losing game into an apparent winner.
The long-run average
Expected value multiplies each outcome by its probability and sums. It is what you would average over very many repetitions, not what you should expect on any single trial.
It need not be a possible outcome
The expected value of a die roll is 3.5, which no roll produces. An expected value of 2.3 children is meaningful as an average even though no family has 2.3 children.
A fair game has expected value zero
Every commercial gambling game has a negative expected value for the player — that is what makes it a business. Computing the expected value of a bet is the quickest way to see the house edge.
And where it is a bad guide
Expected value ignores risk. A bet with positive expected value that could bankrupt you is still a bad bet. Insurance has negative expected value for the buyer and is still rational, for the same reason.
Step 2: Try It Yourself
Tap and try it out.
Win 10
Lose 2
Lose 2 has the most. It has 8 more than Win 10.
Step 3: Watch an Example
One step at a time.
Watch Yusuf Judge a Game
A game costs $2. It pays $10 with probability 0.1 and nothing otherwise.
- Step 1
The expected payout is 10 × 0.1 + 0 × 0.9, which is $1.
Step 4: Your Turn
Practice makes it stick.
The Payout
Problem 1 of 2
A game pays $20 with probability 0.25 and $0 otherwise. What is the expected payout, in dollars?
dollars
The Die
Problem 2 of 2
What is the expected value of one roll of a fair six-sided die?
The Long Run
1 of 8
Pays $10 with probability 0.5, else $0. Expected payout in dollars?
2 of 8
Pays $100 with probability 0.01, else $0. Expected payout?
3 of 8
Expected payout $5, cost to play $5. Expected gain?
4 of 8
Is that game fair? 1 for yes, 0 for no.
5 of 8
Expected payout $3, cost $4. Expected gain in dollars?
6 of 8
Pays $6 with probability 0.5 and $2 with probability 0.5. Expected payout?
7 of 8
Which statements about expected value are true?
8 of 8
Pays $50 with probability 0.02, else $0. Expected payout?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Pays $40 with probability 0.1, else $0. Expected payout in dollars?
Question 2 of 2
What makes a game fair?
What You Learned
- Expected value multiplies each outcome by its probability and adds.
- It is a long-run average and may never occur as a single result.
- A fair game has expected gain zero, once the cost of playing is subtracted.