Understanding Expected Value: Definition, Examples, and Applications

Expected value is a fundamental concept in probability, finance, and decision-making. Whether you’re evaluating investments, playing poker, or assessing business risks, understanding expected value helps you make rational choices. In this article, I break down what expected value means, how to calculate it, and where it applies in real-world scenarios.

What Is Expected Value?

Expected value (EV) represents the long-term average outcome of a random event if repeated many times. It weighs all possible outcomes by their probabilities, providing a single summary measure. Mathematically, for a discrete random variable $X$ with possible outcomes $x_1, x_2, \dots, x_n$ and probabilities $P(x_1), P(x_2), \dots, P(x_n)$, the expected value is:

$E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)$

For continuous variables, we use integration:

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

where $f(x)$ is the probability density function.

A Simple Example: Rolling a Die

Consider a fair six-sided die. Each outcome (1 through 6) has a probability of $\frac{1}{6}$. The expected value is:

$E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + \dots + 6 \cdot \frac{1}{6} = 3.5$

This means, over many rolls, the average value converges to 3.5, even though 3.5 itself is not a possible outcome in a single roll.

Expected Value in Finance and Investing

Investors use expected value to assess potential returns. Suppose you have two investment options:

1. Stock A: 60% chance of a 10% return, 40% chance of a -5% loss.
2. Stock B: 80% chance of a 5% return, 20% chance of a -2% loss.

Calculating EV for both:

$E(A) = 0.6 \times 10 + 0.4 \times (-5) = 6 - 2 = 4\%$

$E(B) = 0.8 \times 5 + 0.2 \times (-2) = 4 - 0.4 = 3.6\%$

Stock A has a higher expected return, but it also carries more risk. Expected value alone doesn’t capture volatility, so investors often combine it with variance or standard deviation.

Comparing Investment Strategies

Strategy	Probability of Gain	Gain (%)	Probability of Loss	Loss (%)	Expected Value (%)
Aggressive	60%	15	40%	-10	5
Moderate	75%	8	25%	-3	5.25
Conservative	90%	4	10%	-1	3.5

Here, the moderate strategy has the highest EV, but risk-averse investors might prefer the conservative approach.

Expected Value in Gambling and Decision-Making

Casino games are designed with negative expected values for players, ensuring house profitability. Take roulette:

• American Roulette: 38 slots (1-36, 0, 00). Betting $1 on a single number pays $35.
• Probability of winning: $\frac{1}{38}$
• Probability of losing: $\frac{37}{38}$

The expected value is:

$E(X) = 35 \cdot \frac{1}{38} + (-1) \cdot \frac{37}{38} = -0.0526$

This means, per $1 bet, you lose about 5.26 cents on average.

Poker and Expected Value

In poker, calculating EV helps decide whether to call, fold, or raise. Suppose:

• Pot size: $100
• Opponent bets $20
• You estimate a 25% chance of winning

The EV of calling is:

$E(X) = 0.25 \times 120 + 0.75 \times (-20) = 30 - 15 = 15$

A positive EV suggests calling is profitable in the long run.

Business Applications: Risk Assessment

Businesses use expected value to evaluate projects. Suppose a company considers launching a product:

• Development cost: $500,000
• 70% chance of success (profit: $1,000,000)
• 30% chance of failure (profit: $0)

The expected net gain is:

$E(X) = 0.7 \times (1,000,000 - 500,000) + 0.3 \times (-500,000) = 350,000 - 150,000 = 200,000$

A positive EV supports the decision to proceed.

Expected Value vs. Utility Theory

While EV is mathematically sound, humans don’t always follow it due to risk aversion. Losing $1,000 hurts more than gaining $1,000 feels good. Daniel Bernoulli introduced utility theory to account for this, suggesting decisions depend on subjective value rather than pure monetary outcomes.

Limitations of Expected Value

1. Ignores Variance: Two investments with the same EV may have different risk levels.
2. Assumes Repeatability: EV works best for repeated events, not one-time decisions.
3. Probability Estimation Errors: Garbage in, garbage out—if probabilities are wrong, EV is misleading.

Conclusion

Expected value is a powerful tool for rational decision-making in finance, gambling, and business. By quantifying average outcomes, it helps compare choices objectively. However, it’s not a standalone solution—risk tolerance, utility, and context matter. Mastering EV means balancing math with real-world judgment.