Expected value

Japanese school year: Math A

What you learn

This topic covers calculating the expected value by summing products of numerical outcomes and their probabilities. It is extensively applied in risk assessment, lottery evaluations, and insurance pricing to support decision-making. Students should be familiar with calculating basic probabilities and adding values before studying this concept.

Go to practice

Key points

This formula calculates the expected value $E$ (the theoretical average). Multiply each possible value $x_k$ by its probability $p_k$ , and sum them across all possible outcomes.

E = \sum_{k=1}^n x_k p_k

This fundamental property states that the sum of probabilities for all possible outcomes must always equal 1 ( $100\%$ ). Use it to verify probability distribution tables or find unknown probabilities.

\sum_{k=1}^n p_k = 1

This formula finds the new expected value when a variable $X$ is scaled by $a$ and shifted by $b$ . Simply multiply the original expected value $E(X)$ by $a$ and add $b$ .

E(aX + b) = aE(X) + b

Choose a set to practice.

Practice 1Coming soon