Statistics From Zero · Lesson 13 of 22
Expected Value: What a Bet Is Worth on Average
In one sentence: Expected value is what a choice pays on average if you could repeat it many times; it explains why casinos always win, why insurance can still be sensible, and why a high win rate can lose money.
The idea
Every choice with uncertain results, from a bet to an insurance policy to a trade, can be summarised by one number: what it pays on average if you could make the same choice again and again. That number is the expected value.
You work it out by taking each possible outcome, multiplying it by its probability, and adding them up. That's all. The name is misleading: the expected value is often a result you can't actually get on any single go. It's what you'd average over many goes.
Uncle Gary at the fair
At the summer fair, a stall offers a game: pay $2, roll a die, and if you roll a six you win $10. Uncle Gary has rolled 9 times. He won once, and he describes the game as "nearly paying for itself".
Let's value one go:
- 1 in 6 chance of winning $10: worth 10 × 1/6 = $1.67 on average.
- You always pay $2.
- Expected value: $1.67 - $2 = -$0.33 per go.
On average, every roll costs Gary 33 cents. He'll win sometimes, and the wins will feel great, but over many rolls the stall collects about a sixth of everything he hands over. That's the stall's house edge, and it's how every casino, lottery and fairground game is built: some players win, the average player loses, and the house is the average player's opponent forever. (Our lottery article is the extreme case.)
When the average isn't everything: insurance
Aunt Linda pays $8 a month to insure her $120 phone. Say there's a 1 in 20 chance of breaking it in a given month. The expected loss is 120 × 1/20 = $6 a month, so the insurance has an expected value of -$2 a month. By the numbers alone, it's a bad bet, and the insurer is counting on that.
Is Linda wrong? Not necessarily. Expected value tells you the average, not the risk. If losing $120 at once would be a real problem this month, paying a small, certain $8 to avoid a rare, painful $120 can be sensible. That's what insurance is for: trading a slightly worse average for a much smaller worst case. It stops being sensible when you insure things you could easily replace, again and again, and pay the house edge every time.
So: expected value is the right guide when you'll repeat a choice many times and can survive the bad outcomes. When one bad outcome would hurt badly, the spread of outcomes (Lesson 3) matters too.
The trap of the high win rate
Cousin Kevin has a trading strategy that wins 60% of the time. He tells everyone. He doesn't tell anyone that his wins average +1% and his losses average -2%.
Expected value per trade: 0.6 × (+1%) + 0.4 × (-2%) = 0.6% - 0.8% = -0.2%. Kevin wins most of the time and loses money overall. A win rate on its own tells you nothing; you need the size of the wins and losses too. It's also why a tight stop-loss can produce lots of small losses and a worse average, which we found in our stop-loss test.
Try it yourself
Gary is offered two prizes in a pub quiz final. A: a guaranteed $45. B: a coin toss, heads $100, tails nothing.
1. What's the expected value of each? 2. Which should he choose? 3. Would your answer change if he could play this exact choice 1,000 times?
Show the answer
1. A is worth $45. B is worth 0.5 × $100 + 0.5 × $0 = $50.
2. B has the higher average. But it's one go: half the time Gary walks away with nothing. If $45 for certain matters more to him than a coin toss for $100, taking A is perfectly reasonable. Choosing the safer option at a small cost to the average is called being risk-averse, and most people are, sensibly.
3. Over 1,000 goes, B would almost certainly pay close to $50,000 against A's $45,000. With many repeats, the average wins and the risk mostly washes out. That's exactly why casinos love expected value and individual gamblers shouldn't trust their luck.
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Expected value, E(X) | The probability-weighted average of all possible outcomes |
| Random variable | A number whose value depends on chance (the payout of one roll) |
| House edge | The expected loss per bet for the player, built into the game |
| Fair game | A bet whose expected value is exactly zero |
| Variance (of outcomes) | How spread out the possible results are; a measure of risk |
| Risk-averse | Willing to accept a lower average for less uncertainty |
| Win rate (hit rate) | The share of attempts that win; meaningless without the size of wins and losses |
Three things to remember
- Expected value = each outcome times its probability, added up. It's the average over many goes.
- Over many repeats the average wins; for a single, painful outcome, risk matters too.
- A high win rate means nothing without the size of the wins and the losses.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.