Expected Value Calculator

The expected value ($E(X)$) represents the weighted average of all possible numerical outcomes, where each outcome is multiplied by its respective probability of occurrence. For a discrete random variable $X$ with possible values $x_1, x_2, ..., x_n$ and corresponding probabilities $p_1, p_2, ..., p_n$, the formula is expressed as:

$E(X) = \sum_{i=1}^{n} x_i \cdot p_i = x_1p_1 + x_2p_2 + ... + x_np_n$

A fundamental rule of this calculation is that the sum of all individual probabilities must equal exactly 1 (or 100%). If your probabilities do not sum to 1, the model represents an invalid probability distribution and requires normalization.

Imagine you are evaluating a small business investment that has three distinct financial outcomes over a one-year period:

1. A 20% ($0.20$) chance to earn a profit of $10,000.
2. A 50% ($0.50$) chance to break even ($0 profit).
3. A 30% ($0.30$) chance to lose $4,000.

To find the expected return, we multiply each outcome by its probability and sum the results:

$E(X) = (10,000 \cdot 0.20) + (0 \cdot 0.50) + (-4,000 \cdot 0.30)$

$E(X) = 2,000 + 0 - 1,200 = 800$

The expected value of this investment is $800, indicating that over many repeated trials under identical conditions, you would average an $800 gain per venture.

Always verify that your probability inputs sum to 1.0 before finalizing your analysis; failing to account for all possible scenarios will skew your mathematical expectation. Additionally, remember that expected value is a theoretical long-term average. In a single trial, you will almost never experience the exact expected value—especially in high-variance scenarios.