Expected Value Calculator

What Is Expected Value?

Expected value (EV) is the long-run average outcome of a random variable when an experiment is repeated many times. It represents the "center of mass" of a probability distribution and is one of the most important concepts in probability theory and decision-making.

Despite its name, the expected value isn't necessarily a value you'd "expect" to see in any single trial—it's the average you'd approach over many repetitions.

The Formula

For a discrete random variable with outcomes $x_i$ and probabilities $P(x_i)$:

$E[X] = \sum_{i} x_i \cdot P(x_i)$

For a continuous random variable with probability density function $f(x)$:

$E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$

A Simple Example

Consider a fair six-sided die. The expected value is:

$E[X] = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} = 3.5$

You can never roll a 3.5, yet this is the expected value. If you rolled the die thousands of times, your average would converge to 3.5.

Practical Applications

Gambling and Casino Games

Casinos ensure every game has a negative expected value for players. In roulette, betting $1 on red:

• Win $1 with probability 18/38
• Lose $1 with probability 20/38

$E[X] = (1)\frac{18}{38} + (-1)\frac{20}{38} = -\frac{2}{38} \approx -\$0.053$

The house edge of about 5.3 cents per dollar bet ensures casino profitability over millions of bets.

Insurance

Insurance companies use expected value to price policies. If there's a 1% chance of a $100,000 claim, the expected payout is$1,000. The premium must exceed this to be profitable.

Investment Decisions

Portfolio managers calculate expected returns by weighting possible outcomes by their probabilities:

$E[Return] = (0.20)(0.30) + (0.08)(0.50) + (-0.15)(0.20) = 0.07 = 7\%$

Business Decisions

Expected value helps compare options with uncertain outcomes. Should a company pursue Project A with uncertain payoffs or Project B with a guaranteed return?

Properties of Expected Value

Linearity

For any random variables X and Y and constants a, b:

$E[aX + bY] = aE[X] + bE[Y]$

This property is extraordinarily useful and holds regardless of whether X and Y are independent.

Expected Value of a Constant

$E[c] = c$

Transformation

If $Y = g(X)$, then:

$E[Y] = E[g(X)] = \sum_{i} g(x_i) \cdot P(x_i)$

Expected Value vs. Most Likely Outcome

The expected value differs from the mode (most likely outcome). In a lottery where:

• 99.9% chance of winning $0
• 0.1% chance of winning $10,000

The mode is 0, but: $$E[X] = (0)(0.999) + (10000)(0.001) = \10$$

Variance and Expected Value

The variance measures spread around the expected value:

$Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$

Two investments might have the same expected return but very different variances—a critical consideration for risk-averse investors.

The Law of Large Numbers

The Law of Large Numbers guarantees that as the number of trials increases, the sample average converges to the expected value. This is why:

• Casinos are profitable despite individual gamblers occasionally winning
• Insurance companies can reliably estimate payouts
• Quality control sampling works

Decision Theory

Expected value is central to decision theory, but it's not the whole story. Consider:

• A 50% chance of winning $1,000,000 vs. a guaranteed$400,000

The expected values are $500,000 and$400,000, but most people prefer the guaranteed amount. This leads to expected utility theory, which accounts for risk preferences.

Related Calculations

• Variance Calculator — Measure spread around expected value
• Standard Deviation Calculator — Square root of variance
• Binomial Probability Calculator — Expected value is $np$
• Poisson Probability Calculator — Expected value equals $\lambda$
• Weighted Average Calculator — Similar weighted calculation

Summary

Expected value provides a single number summarizing a random variable's long-run average behavior. It's foundational to probability, statistics, economics, and decision-making. While indispensable, expected value alone doesn't capture risk—always consider variance alongside it for a complete picture of uncertain outcomes.