Mathematics, Statistics & Geometry

Expected Value Calculator

Calculate the expected value E(X) and variance of a discrete probability distribution. Essential for risk analysis and statistical modeling.

Expected Value E(X)
26
Variance Var(X)84
Standard Deviation9.165
Calculation StepsFormula: E(X) = Σ (x * p(x)) Term 1: 10 * 0.1000 = 1.0000 Term 2: 20 * 0.4000 = 8.0000 Term 3: 30 * 0.3000 = 9.0000 Term 4: 40 * 0.2000 = 8.0000 Total Expected Value (Mean) = 26.000000

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Calculating Risk and Reward

The Expected Value Calculator is the most powerful mathematical tool in a decision-maker's arsenal. By weighting potential payouts against their mathematical likelihoods, it reveals the true long-term value of any statistical bet or investment.

E(X)=∑i=1nxipi\begin{aligned} E(X) = \sum_{i=1}^{n} x_i p_i \end{aligned}

Where:
E(X)=
The long-term average outcome
xix_i=
A specific possible numerical outcome
pip_i=
The chance of that specific outcome occurring

The Law of Large Numbers

Expected value is built upon the Law of Large Numbers, which states that as the number of trials increases, the actual observed average will converge on the mathematical expected value. Short-term variance (luck) disappears over the long term.

Real-World Applications

  • Casino Mathematics: Every game in a casino has a negative expected value for the player. Even if a player wins in the short term, the casino's positive expected value guarantees long-term profit.
  • Insurance Underwriting: Actuaries calculate the expected value of potential claims (probability of an accident ×\times cost of the accident) to set profitable monthly premiums.
  • Stock Market Trading: Quantitative analysts use expected value to evaluate the mathematical edge of various algorithmic trading strategies.

Frequently Asked Questions

Expected Value is the predicted average outcome of a random event if you were to repeat it an infinite number of times.

Often, no. For example, the expected value of rolling a 6-sided die is 3.5. You cannot actually roll a 3.5, but if you roll the die 1000 times, the average of all your rolls will be very close to 3.5.

In statistics, valid probabilities must sum to 1 (100%). If your inputs do not sum to 1, this calculator automatically normalizes them, treating them as relative frequencies or 'weights'.

A regular mean assumes all outcomes are equally likely. Expected value is a 'weighted mean' that gives more mathematical importance to outcomes that have a higher chance of occurring.

Variance and standard deviation measure 'risk' or 'volatility'. Two games can have the same expected value, but if one has much higher variance, the actual outcomes will swing much more wildly.

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