Expected Value

Learn how to calculate the average outcome of a random event over many trials.

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Definition

The expected value (also called the mean or expectation) of a random variable is the long-run average value you expect to get if you repeat an experiment many times.
For a discrete random variable XX with outcomes x1,x2,…,xnx_1, x_2, \ldots, x_n and corresponding probabilities P(x1),P(x2),…,P(xn)P(x_1), P(x_2), \ldots, P(x_n):
E(X)=x1⋅P(x1)+x2⋅P(x2)+⋯+xn⋅P(xn)E(X) = x_1 \cdot P(x_1) + x_2 \cdot P(x_2) + \cdots + x_n \cdot P(x_n)
Or more compactly:
E(X)=∑i=1nxi⋅P(xi)E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)
Key insight: Expected value is a weighted average where each outcome is weighted by its probability.

Try it now

What does the expected value of a random experiment represent?

Worked Examples

What is the expected value when rolling a fair six-sided die?
1

List all outcomes and their probabilities

Outcomes: 1,2,3,4,5,61, 2, 3, 4, 5, 6. Each has probability 16\frac{1}{6} → 6 equally likely outcomes
2

Apply the expected value formula

E(X)=1⋅16+2⋅16+3⋅16+4⋅16+5⋅16+6⋅16E(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} → Sum of weighted outcomes
3

Simplify the calculation

E(X)=16(1+2+3+4+5+6)=216E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} → 216=3.5\frac{21}{6} = 3.5

Common Mistakes

Forgetting that expected value can be a non-integer (like 3.5 on a die)

Why it's wrong: Expected value is a theoretical average, not an actual outcome. You cannot roll 3.5, but over many rolls, your average will approach 3.5.

Correct: Understand that expected value represents the long-run average, not a specific outcome you will get.

Using equal probabilities when outcomes are not equally likely

Why it's wrong: A biased coin or weighted die has different probabilities for each outcome.

Correct: Always check if outcomes are equally likely. Use the actual probability for each outcome.

Forgetting to subtract costs when calculating expected profit

Why it's wrong: Winning 10 euros is not profit if you paid 15 euros to play!

Correct: E(profit)=E(winnings)−costE(\text{profit}) = E(\text{winnings}) - \text{cost}

Adding probabilities instead of multiplying outcome by probability

Why it's wrong: The formula is x⋅P(x)x \cdot P(x), not x+P(x)x + P(x).

Correct: Each outcome must be multiplied by its probability, then sum all the products.

Interactive Visual

Dice Roller

Click the button to run trials and see probability in action!

Dot Plot

Die faces: the mean is E(X) = 3.5

123456X

Each dot is one value (6 values in total).

Mean = 3.5

Tap a column to count its dots.

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Practice Problems

16 problems
Problem 1 of 16
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What does the expected value of a random experiment represent?

Why It Matters

Expected value is one of the most powerful concepts in probability and statistics:
  • Casino games: Casinos set payouts so that the expected value favors the house
  • Insurance: Companies calculate expected payouts to set premium prices
  • Investments: Investors compare expected returns to make decisions
  • Game theory: Expected value helps determine optimal strategies
  • Quality control: Manufacturers predict average defect rates
Understanding expected value helps you make rational decisions when outcomes are uncertain!

Real World Applications

Casino Games and House Edge

Casinos design games so that the expected value always favors the house. In roulette, the house edge means players lose about 2.7% (European) or 5.3% (American) of their bets on average.

Example:

In European roulette, betting 10 euros on red gives: E=10⋅1837+(−10)⋅1937≈−0.27E = 10 \cdot \frac{18}{37} + (-10) \cdot \frac{19}{37} \approx -0.27 euros. You lose 27 cents on average per 10-euro bet.

1Try It Yourself

A game pays 35 times your bet if you guess the exact number on a roulette wheel (37 numbers total). You bet 1 euro.

What is the expected value of this bet?

Step 1: Write the mathematical expression

Calculate: E=35⋅137+(−1)⋅3637E = 35 \cdot \frac{1}{37} + (-1) \cdot \frac{36}{37}

Investment Decisions

Investors use expected value to compare different investment options by weighing potential returns against their probabilities.

Example:

Stock A has a 60% chance of gaining 20% and a 40% chance of losing 10%. E(return)=0.20⋅0.60+(−0.10)⋅0.40=0.08=8%E(\text{return}) = 0.20 \cdot 0.60 + (-0.10) \cdot 0.40 = 0.08 = 8\% expected return.

2Try It Yourself

Investment B has a 30% chance of gaining 50% and a 70% chance of gaining 5%.

What is the expected return of Investment B?

Step 1: Write the mathematical expression

Calculate: E=0.50⋅0.30+0.05⋅0.70E = 0.50 \cdot 0.30 + 0.05 \cdot 0.70

Game Show Strategy

Game show contestants use expected value to decide whether to keep a guaranteed prize or risk it for a chance at something bigger.

Example:

You have won 1000 euros. You can keep it or flip a coin: heads wins 2500 euros, tails wins nothing. E(flip)=2500⋅0.5+0⋅0.5=1250E(\text{flip}) = 2500 \cdot 0.5 + 0 \cdot 0.5 = 1250 euros. Mathematically, you should flip!

3Try It Yourself

You won 500 euros. You can keep it OR spin a wheel: 25% chance of 3000 euros, 75% chance of 0 euros.

What should you do based on expected value?

Step 1: Write the mathematical expression

Calculate: E(spin)=3000⋅0.25+0⋅0.75E(\text{spin}) = 3000 \cdot 0.25 + 0 \cdot 0.75

Key Takeaways

  • 1Expected value E(X)E(X) is the long-run average of a random experiment repeated many times
  • 2Formula: E(X)=∑xi⋅P(xi)E(X) = \sum x_i \cdot P(x_i) (sum of each outcome times its probability)
  • 3Expected value can be a number you cannot actually get (like 3.5 on a die)
  • 4Positive expected value means you gain on average; negative means you lose
  • 5To find expected profit, subtract costs from expected winnings

Frequently Asked Questions

Expected value is the average over many trials, not a single outcome. If you roll a die 1000 times and average all results, you'll get very close to 3.5. It's a theoretical center of the distribution.
Expected value is the average over many trials, not a single outcome. If you roll a die 1000 times and average all results, you'll get very close to 3.5. It's a theoretical center of the distribution.
Expected value (mean) weights outcomes by probability. The mode is the most likely outcome, and the median is the middle value. For a fair die: mean = 3.5, mode = none (all equally likely), median = 3.5.
Yes! If losses outweigh gains on average, the expected value is negative. Most gambling games have negative expected value for players, which is why casinos are profitable.
Not necessarily. Expected value ignores risk tolerance. Would you bet your entire savings on a coin flip that doubles or loses it all? The expected value is positive, but most people wouldn't take that risk.

Glossary

Expected value
The long-run average outcome of a random experiment, calculated as the sum of each outcome times its probability. Notation: E(X)E(X)
Random variable
A variable whose value depends on the outcome of a random event
Probability distribution
A list of all possible outcomes and their probabilities
Weighted average
An average where some values contribute more than others, based on their weights (here, probabilities)
House edge
The casino's expected profit as a percentage of each bet
Today's challengeOne problem, three tries

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