Quick Answer: Expected Value (EV) is the average outcome you would get if you repeated a risky decision thousands of times. It is calculated by multiplying each possible outcome by its probability and adding them together. A positive EV means a good bet; a negative EV means a bad bet.
The Expected Value Formula
EV = (Probability of Outcome A × Value of Outcome A) + (Probability of Outcome B × Value of Outcome B) ... and so on.
A Simple Game: The Coin Flip Bet
Imagine a simple game: we flip a fair coin. If it is heads, you win $10. If it is tails, you lose $5. Should you play? Let's calculate the Expected Value:
- Outcome 1 (Heads): 50% probability × +$10 = +$5.00
- Outcome 2 (Tails): 50% probability × -$5 = -$2.50
- Total EV: $5.00 + (-$2.50) = +$2.50
The EV is +$2.50. This means that, on average, you will make $2.50 every time you play this game. You might lose on the first flip, but over hundreds of flips, you are guaranteed to profit. You should play.
Why You Shouldn't Play the Lottery
Let's calculate the EV of a typical $2 lottery ticket with a 1-in-300-million chance to win $100 million.
- Winning outcome: (1/300,000,000) × $100,000,000 = +$0.33
- Losing outcome: (299,999,999/300,000,000) × -$2.00 ≈ -$2.00
- Total EV ≈ -$1.67
For every $2 ticket you buy, your expected result is losing $1.67. This mathematically proves why lotteries are "a tax on people who are bad at math."
Expected Value in Business
Entrepreneurs, investors, and poker players all use EV. "There is a 20% chance this marketing campaign makes $50k, and an 80% chance it loses $5k." The EV is ($50k × 0.2) + (-$5k × 0.8) = $10k - $4k = +$6k. The campaign is a mathematically sound investment, even though it fails 80% of the time.