🧬 kbird.ai

Expected Value

4 min read

Expected Value

Probability × Payoff

Definition

Expected value (EV) is the sum of all possible values of a random variable, each multiplied by its probability of occurrence. It represents the average outcome if an action were repeated many times.

EV = (Probability of Outcome 1 × Value of Outcome 1) +
     (Probability of Outcome 2 × Value of Outcome 2) +
     ...

“Take the probability of loss times the amount of possible loss from the probability of gain times the amount of possible gain.” — Blaise Pascal (Pascal’s Wager)

Simple Examples

Coin Flip Game

  • Heads: Win $10 (probability 0.5)
  • Tails: Lose $5 (probability 0.5)
EV = (0.5 × $10) + (0.5 × -$5) = $5 - $2.50 = $2.50

Interpretation: Play this game 100 times, expect to gain ~$250.

Lottery Ticket

  • Cost: $2
  • Win $1 million: Probability 1 in 10 million
  • Win nothing: Probability 9,999,999 in 10 million
EV = (0.0000001 × $999,998) + (0.9999999 × -$2)
EV = $0.10 - $2.00 = -$1.90

Interpretation: For every 1.90.

Business Decision

Launch a product:

  • 30% chance of $1M profit
  • 40% chance of $100k profit
  • 30% chance of $200k loss
EV = (0.3 × $1M) + (0.4 × $100k) + (0.3 × -$200k)
EV = $300k + $40k - $60k = $280k

Interpretation: Average expected profit is $280k.

When to Use Expected Value

Good Uses

  • Repeated decisions (investing, business)
  • Portfolio decisions (many bets)
  • Risk quantification
  • Comparing options mathematically

Poor Uses

  • One-time irreversible decisions
  • Outcomes where magnitude matters more than frequency
  • Situations with catastrophic tail risks
  • When utility isn’t linear with money

Beyond Money: Utility

Diminishing Marginal Utility

  • 0 than to a billionaire
  • Losing 10,000 feels good (loss aversion)

Expected Utility

Replace dollar values with “utility” (subjective value):

  • $1M might = 500 utils for a poor person
  • $1M might = 10 utils for a billionaire

Same EV in dollars, different expected utility.

The Kelly Criterion

How much to bet when you have an edge?

f* = (bp - q) / b

Where:
f* = fraction of bankroll to bet
b = odds received (e.g., bet $1 to win $b)
p = probability of winning
q = probability of losing (1 - p)

Key insight: Even with positive EV, bet too much and you go broke (gambler’s ruin).

Common Mistakes

Ignoring Variance

Two investments with same EV:

  • A: 50% chance of 0 (EV = $50)
  • B: 100% chance of 50)

A has higher variance. Risk-averse people prefer B.

Overconfidence in Probabilities

EV is only as good as probability estimates. We often:

  • Overestimate our knowledge
  • Ignore unknown unknowns
  • Use point estimates when ranges are appropriate

Tail Risks

Low probability × catastrophic outcome = negative EV we ignore

Example: “It probably won’t happen” × “Company goes bankrupt” = Don’t do it

In Real Decisions

When EV Is Positive

You should generally take positive EV bets, but consider:

  • Can you afford the loss?
  • Is it repeatable?
  • Are you calibrated on probabilities?

When EV Is Negative

Avoid negative EV in repeated games:

  • Casinos (house edge)
  • Lottery (state takes 40%+)
  • Insurance (company must profit)

Exception: Insurance has negative EV in dollars but positive EV in utility (avoiding ruin).

References

  • von Neumann, J. & Morgenstern, O. (1944). Theory of Games and Economic Behavior
  • Thorp, E.O. (2006). The Kelly Capital Growth Investment Criterion
  • Mauboussin, M. (2012). The Success Equation

Calculate carefully, bet wisely, repeat often. 🎲

Graph View

Backlinks