1. What is Expected Value?

The expected value is a fundamental concept in probability that represents the average outcome if an experiment is repeated many times. It's calculated by summing the products of each possible value and its probability of occurrence.

2. How Does the Calculator Work?

The calculator uses the expected value formula:

Where:

• \( E[X] \) — Expected value (same units as x_i)
• \( x_i \) — Possible values of the random variable
• \( P(x_i) \) — Probability of each value (0 ≤ P ≤ 1)

Explanation: The formula multiplies each possible outcome by its probability and sums all these products to get the long-run average.

3. Importance of Expected Value

Details: Expected value is crucial in decision making, risk assessment, statistics, and many real-world applications like insurance, finance, and game theory.

4. Using the Calculator

Tips: Enter one value per line in the first box and its corresponding probability (between 0 and 1) in the second box. The sum of all probabilities should equal 1 for proper probability distributions.

5. Frequently Asked Questions (FAQ)

Q1: What if my probabilities don't sum to 1?
A: For a proper probability distribution, probabilities should sum to 1. If they don't, the result won't represent a true expected value.

Q2: Can I use percentages instead of decimals?
A: Yes, but convert them to decimals (e.g., 30% = 0.3) before entering.

Q3: What's the difference between expected value and average?
A: They're conceptually similar, but expected value is theoretical (based on probabilities) while average is empirical (based on observed data).

Q4: Can expected value be negative?
A: Yes, if some x_i values are negative and have sufficient probability.

Q5: How is expected value used in real life?
A: Applications include insurance premiums, investment decisions, game strategies, and quality control processes.