Expected Value
Expected value is the sum of possible outcomes weighted by their probabilities. A lottery example shows why a higher chance of winning a prize can still come with an expected financial loss.
THE IDEA
Expected Value
Expected value (EV) is the mathematical framework that quantifies the average outcome of a decision involving risk.
The Lottery Argument
The classic argument against lotteries is that they are a "tax on the stupid". Adam Smith argued that buying more tickets makes you more likely to be a loser.
Consider a lottery with distinct ticket combinations:
- If there are 10 million combinations and 1 winner, buying 1 ticket gives a 9,999,999/10m chance of losing.
- Buying 2 tickets lowers the chance of losing.
- Buying 6 million tickets gives a 60% chance of winning, and only a 40% chance of losing. Smith was wrong that buying more tickets increases your chance of losing.
- However, buying 6 million tickets may cost as much as the entire jackpot. Winning a prize is different from making a net profit.
This is where expected value clarifies the picture.
The EV Calculation
EV = (Probability of Outcome A × Value of A) + (Probability of Outcome B × Value of B)
For the single lottery ticket: (9,999,999/10,000,000 × $0) + (1/10,000,000 × $6,000,000) = $0.60
The expected gross payout is $0.60, while the ticket costs $1.00. On these assumptions, the expected net return is negative.
Why it Matters
Expected value incorporates both the probability of an event and its magnitude. Simple heuristics like "buying more tickets makes you more likely to lose" ignore the magnitude. EV gives different gambles a common scale, though an average alone does not describe how large a loss someone can afford.
When to use it
Evaluating whether to buy an extended warranty or insurance policy.
Comparing job offers with different salary structures, bonuses, and security levels.
Deciding on an investment portfolio based on historical returns, risk, and diversification.
Assessing a business strategy by calculating the expected return on investment (ROI) for different marketing campaigns or product launches.
FROM THEORY TO EVERYDAY LIFE
In practice
The Lottery Calculation
A lottery with 10 million combinations, a $1 ticket, and a $6 million jackpot has an expected value of $0.60. (9,999,999/10,000,000 × $0) + (1/10,000,000 × $6,000,000) = $0.60. This shows the ticket is a losing proposition in purely financial terms.
The Multiple Ticket Fallacy
Buying more distinct tickets reduces the chance of winning no prize. Buying 6 million tickets gives a 60% chance of winning the jackpot and breaking even under this example’s assumptions, and a 40% chance of losing the entire ticket cost. A greater chance of a winning ticket does not imply a positive expected profit.
A MOMENT TO REFLECT
Put it to work.
Think of a major decision you've made (e.g., starting a business, moving cities, a large purchase). What were the probable outcomes and their values? How does framing it as an expected value calculation change your perspective on the decision's quality, regardless of the actual outcome?
Adam Smith's intuition about lotteries was flawed because he confused the probability of losing with the magnitude of the loss. Can you identify a current belief or common saying that might be confusing probability for magnitude, or vice versa?
Expected value treats risk as a purely mathematical calculation. Where do you think the limits of this framework are? When does the 'expected value' of a choice fail to capture its real human value (e.g., playing the lottery for the thrill, insuring against a very rare but catastrophic event, choosing a stable job over a risky startup)?
Often confused with
References
- An Inquiry into the Nature and Causes of the Wealth of Nations — Adam Smith (1776)
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)