How to Calculate Expected Value: A Step-by-Step Manual Guide

What You'll Need to Calculate Expected Value by Hand

To compute expected value manually, you only need a list of possible outcomes and their probabilities. No special software is required – just a calculator or paper and pen. This guide walks through the process step by step, using clear examples.

Step-by-Step Process

Follow these steps to find the expected value of any probability distribution.

1. List all possible outcomes – Identify every distinct result of the random event. For example, a single die has outcomes 1, 2, 3, 4, 5, and 6.
2. Assign probabilities – Determine the probability P(x) for each outcome. Probabilities must be between 0 and 1 and add up to 1 (or 100%). If they don't, check your values.
3. Multiply each outcome by its probability – For each outcome x, compute x × P(x). This gives the weighted contribution of that outcome.
4. Sum all the products – Add together all the values from step 3. The result is the expected value E[X].
5. Verify your work – Confirm that probabilities sum to 1 and that the result makes sense given the context.

For more on the formula behind this process, see our Expected Value Formula page.

Worked Example 1: Single Die Roll

Scenario: Roll a fair six-sided die. What is the expected value of the number shown?

The outcomes are 1, 2, 3, 4, 5, 6, each with probability 1/6.

• 1 × 1/6 = 1/6 ≈ 0.1667
• 2 × 1/6 = 2/6 ≈ 0.3333
• 3 × 1/6 = 3/6 = 0.5
• 4 × 1/6 = 4/6 ≈ 0.6667
• 5 × 1/6 = 5/6 ≈ 0.8333
• 6 × 1/6 = 6/6 = 1

Sum: (1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6) = 21/6 = 3.5

Expected value: 3.5. Over many rolls, the average will be close to 3.5.

Worked Example 2: Simple Betting Game

Scenario: A game costs $1 to play. You win $10 with probability 0.4 and lose your $1 with probability 0.6. What is the expected net gain?

Net gain outcomes: Win $9 (because you get $10 minus the $1 cost) or lose $1. Probabilities: P(win) = 0.4, P(lose) = 0.6.

• 9 × 0.4 = 3.6
• (-1) × 0.6 = -0.6

Sum: 3.6 + (-0.6) = 3.0

Expected value: $3.00 per game. On average, you gain $3 each time you play. This is a positive expected value, meaning the game is favorable to you. For more on interpreting such results, see Interpreting Expected Value.

Common Pitfalls

• Probabilities not summing to 1: Always check that total probability equals 1. If it's less, you missed an outcome; if greater, probabilities are too high.
• Confusing probability with frequency: Using raw counts instead of probabilities. For example, if an outcome occurs 3 times out of 10, use 0.3, not 3.
• Forgetting negative outcomes: Losses or negative values must be included with a minus sign. A loss of $5 is -5.
• Misidentifying the outcomes: Be precise about what you're measuring. In the betting game, the outcome is net gain, not total payout.

For a deeper understanding of expected value concepts, read our What Is Expected Value guide.

Practice on Your Own

Now that you know the steps, try calculating expected value for different scenarios – coin flips, dice games, or investment returns. Use our Expected Value Calculator to check your work instantly.