Expected Value
Expected value is the probability-weighted average of possible outcomes. It need not be a possible outcome of any single trial. It can guide pricing, provided the probabilities, costs, and meaning of the payout are specified.
THE IDEA
EXPECTED VALUE IS NOT THE VALUE YOU EXPECT
Expected value is another one of those mathematical notions saddled, like significance, with a name that doesn't quite capture its meaning. We certainly don't "expect" the lottery ticket to be worth 60 cents: on the contrary, it's worth either 6 million clams or zilch, nothing in between.
Similarly: suppose I make a $10 bet on a dog I think has a 10% chance of winning its race. If the dog wins, I get $100; if the dog loses, I get nothing. The expected value of the bet is then
(10% × $100) + (90% × $0)= $10.
But this is not, of course, what I expect to happen. Winning $10 is, in fact, not even a possible outcome of my bet, let alone the expected one. A better name might be "average value"—for what the expected value of the bet really measures is what I'd expect to happen if I made many such bets on many such dogs. Let's say I laid down a thousand $10 bets like that. I'd probably win about a hundred of them (the Law of Large Numbers again!) and make $100 each time, totaling $10,000; so my thousand bets are returning, on average, $10 per bet. In the long run, you're likely to come out about even.
Expected value is a great way to figure out the right price of an object, like a gamble on a dog, whose true value isn't certain. If I pay $12 apiece for those tickets, I'm very likely to lose money in the long run; if I can get them for $8, on the other hand, I should probably buy as many as I can.* Hardly anybody plays the dogs anymore, but the machinery of expected value is the same whether you're pricing race tickets, stock options, lottery tickets, or life insurance.
When to use it
Determining fair price for a lottery ticket
Evaluating whether to buy an extended warranty
Assessing insurance premiums
Deciding whether to invest in a startup with uncertain returns
FROM THEORY TO EVERYDAY LIFE
In practice
Dog Race Bet
A $10 bet on a dog with 10% chance to win $100 has expected value $10 in gross payout. This does not mean you expect to receive $10; you either receive $100 or nothing, with the $10 stake paid either way. Over many comparable bets, average gross payout approaches $10 per bet. Subtracting the stake gives the expected net profit.
Lottery Ticket Pricing
A lottery ticket with 1 in 10 million chance to win $6 million has expected value of $0.60. The ticket is not expected to be worth $0.60; it's either $6 million or $0. The expected value helps determine if the price is fair.
A MOMENT TO REFLECT
Put it to work.
How can expected value help you decide whether to buy a warranty on a new device?
In what real-life decisions have you mistaken expected value for the most likely outcome?
If you have a 1% chance to win $1000 and a 99% chance to lose $1, what is the expected value? Would you take that gamble?
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)