Calculator for Probability and Statistics

Expected Value Calculator

Use this calculator to determine the expected value of a discrete random variable. The expected value represents the long-run average outcome of a random process.

Understanding Expected Value

The expected value (E[X]) of a discrete random variable X is a fundamental concept in probability and statistics. It represents the weighted average of all possible outcomes, where each outcome is weighted by its probability of occurrence. In simpler terms, if you were to repeat a random experiment many times, the expected value is the average outcome you would anticipate.

Formula for Expected Value

For a discrete random variable X with possible outcomes x₁, x₂, …, x_n and their corresponding probabilities P(x₁), P(x₂), …, P(x_n), the expected value is calculated as:

E[X] = Σ [x_i * P(x_i)]

Where:

• x_i is the value of the i-th outcome.
• P(x_i) is the probability of the i-th outcome occurring.
• Σ denotes the sum of all products.

It's crucial that the sum of all probabilities P(x_i) equals 1 (or 100%).

Why is Expected Value Important?

Expected value has wide-ranging applications across various fields:

• Decision Making: In business and finance, it helps evaluate the average outcome of an investment or project, aiding in risk assessment.
• Games of Chance: It can determine if a game is fair, advantageous to the player, or advantageous to the house. A positive expected value means, on average, you'd profit over many plays.
• Insurance: Actuaries use expected value to set premiums, ensuring the company remains profitable while covering claims.
• Statistics: It's a measure of central tendency, providing insight into the "center" of a probability distribution.

Example Calculation

Let's say you're playing a simple game where you roll a fair six-sided die. If you roll a 6, you win $10. If you roll a 1, you lose $5. For any other number (2, 3, 4, 5), you win $1.

Here are the outcomes and their probabilities:

• Outcome 1 (Win $10): Value = 10, Probability = 1/6
• Outcome 2 (Lose $5): Value = -5, Probability = 1/6
• Outcome 3 (Win $1): Value = 1, Probability = 4/6 (for rolling 2, 3, 4, or 5)

Now, let's calculate the expected value:

E[X] = (10 * 1/6) + (-5 * 1/6) + (1 * 4/6)

E[X] = 10/6 – 5/6 + 4/6

E[X] = (10 – 5 + 4) / 6

E[X] = 9 / 6

E[X] = 1.5

The expected value is $1.50. This means that, on average, you would expect to win $1.50 per game if you played many times.

How to Use This Calculator

1. Number of Distinct Outcomes: Enter the total number of different possible results your random variable can have.
2. Enter Outcome Values and Probabilities: For each outcome, input its specific numerical value and its corresponding probability (as a decimal between 0 and 1, or a percentage).
3. Calculate: Click the "Calculate Expected Value" button.
4. View Result: The calculator will display the expected value. It will also warn you if your probabilities do not sum up to 1.

Warning:

" + expectedValue.toFixed(4) + "