Expected value
Weigh each outcome by how likely it is and how much it's worth. Learn when a long shot is a good bet, when a likely win is a bad one, and why a loss you can't survive changes everything.
Expected value (EV)
For each possible outcome, multiply how likely it is by what it's worth to you, then add the results. The total is what the choice is worth on average if you could make it many times. You won't get exactly that number on any single try. It tells you which choices are good ones to keep making.
A coin flip that pays you 100 on heads and costs you 60 on tails: 0.5 × 100 = 50, and 0.5 × −60 = −30. Expected value: +20 per flip. You'll lose about half the flips and still come out ahead over time.
Should Arash buy the extended warranty on his new phone?
Warranty price: 40
Chance of a covered fault in the period: about 5%
Typical repair cost if it happens: 300
Expected repair cost without the warranty:
0.05 x 300 = 15
Compare: pay 40 for certain, or carry an average cost of 15On average the warranty costs Arash 25 more than it returns.
And a 300 repair would annoy him but not break him. He skips it.The numbers are an example. The shape is general: a seller can only offer a warranty at a profit if it costs buyers more than it pays out on average.
Check yourself
Ali has free Saturdays. Option A: tutoring, a guaranteed $100. Option B: a stall at a craft market. The stall fee is $25. On about 6 Saturdays in 10 he'd sell $250, on the other 4 only $50. A bad day wouldn't hurt him. Which reasoning is best?
- A, because B can lose money on a bad day and A never can
- B, because it averages about 145 a Saturday after the fee
- B, because 250 is more than 100 and he only needs a few good days
- They can't be compared, because B is uncertain and A is guaranteed
Show the answer
B, because it averages about 145 a Saturday after the fee
Yes. 0.6 × 250 + 0.4 × 50 = 170, minus the 25 fee is 145, against a sure 100. He'll have bad Saturdays, and over a season he ends up well ahead.
Two long shots
A lottery ticket
Tiny chance, huge prize, small price. It feels like a bargain. But lotteries pay out less than they take in, so the expected value of every ticket is negative. Buy one for fun if you like. It isn't a plan.
Applying for a job you probably won't get
Maybe a 5% chance. The cost is two hours on an application. The payoff is years of better pay and work. Small chance × very large payoff, against a tiny cost: the expected value is strongly positive. Send it.
Check yourself
Shadi pitches her design services to larger clients. About 1 pitch in 8 succeeds, each pitch takes half a day, and a win brings three months of well-paid work. Even though she is turned down most of the time, pitching is a good use of her time.
Show the answer
True
True. Eight pitches cost about four days and bring, on average, one three-month contract. That's a very good trade. Failing 7 times in 8 only sounds bad until you weigh the payoff against the cost. Judge a repeated choice by its average result, not by how often it stings.
THE BIG EXCEPTION
Ruin changes the maths
Expected value is an average over many tries. It only helps you if you're still around for the later tries. If one bad outcome wipes you out, your savings, your health, your home, then the average is a number you never get to collect. So the rule has two parts: prefer positive expected value, and never risk what you can't afford to lose, however good the odds look.
Someone offers a bet: 60% chance to double your life savings, 40% chance to lose them all. On paper the expected value is positive. In real life, 4 times in 10 you are ruined and can't play again. Decline.
Check yourself
Mahsa has modest savings. Which risks should she insure, and which can she carry herself?
- Major hospital costs
- A cracked phone screen
- Injuring someone in a car accident
- A broken toaster after the first year
- Fire in the flat she owns
- Headphones that stop working
Show the answer
Insure it: Major hospital costs, Injuring someone in a car accident, Fire in the flat she owns
Carry it herself: A cracked phone screen, A broken toaster after the first year, Headphones that stop working
Rough expected value in four steps
- List the two or three main outcomes
Good, bad, and maybe middling. You don't need more.
- Give each a rough chance
Use the base rate from the last lesson, not your hopes. '1 in 10' is precise enough.
- Put a rough value on each and multiply
Money, time, or a simple score. Add them up and compare with the alternative.
- Check the worst case on its own
Ask: if the bad outcome happens, am I still standing? If not, make the bet smaller or walk away, whatever the average says.
Check yourself
Sara, a freelancer, is offered two ways to be paid for a project. Fixed: $5,000. Profit share: about a 30% chance of $20,000 and a 70% chance of $1,000. She has six months of living costs saved. What should she weigh?
- The share averages about $6,700, above the fixed fee, and the bad case won't ruin her
- Take the fixed fee: a 70% chance of a bad result settles it, and $5,000 is certain
- Take the share, because $20,000 is four times $5,000
- Expected value can't be used here, since the project happens only once and averages need many repeats
Show the answer
The share averages about $6,700, above the fixed fee, and the bad case won't ruin her
Right. 0.3 × 20,000 + 0.7 × 1,000 = 6,700. It beats 5,000 on average, and with six months saved the bad case hurts without ruining her. Both halves of the rule are satisfied.
Check yourself
- Kian puts 5% of his savings into a friend's risky business: a small chance of a large gain, a large chance of losing it all.
- Dariush puts 90% of his savings into the same business on the same terms.
The odds and payoffs are identical. Why is one a reasonable bet and the other not?
- Kian is luckier than Dariush
- Dariush's expected value is lower per unit of money
- Kian survives the likely bad outcome and can make other bets; Dariush doesn't
- Neither is reasonable, because most risky businesses fail
Show the answer
Kian survives the likely bad outcome and can make other bets; Dariush doesn't
Yes. Same expected value per unit, completely different worst case. The size of the bet relative to what you can afford to lose is part of the decision.
Lesson recap
- Expected value = each outcome's chance × its worth, added up. It's an average over many tries.
- A choice that usually fails can be a great bet if the cost is small and the win is large, and the reverse is also true.
- The average only helps if you survive the bad outcomes, so never risk what you can't afford to lose.
- Insure against ruin and carry small risks yourself.
- Take probabilities from base rates and retest with less kind numbers.