What is the Expected Value?

The expected value is an average outcome that one would expect in case an experiment or action was repeated many times. Our expected value calculator helps you find the average expected outcome from a series of possible results, based on their probabilities. Simply enter the values and their corresponding probabilities to get the expected value.

Here’s an Example

After entering the values & their probabilities provided in the fields, you can check the new fields as you input more values. For Instance:

Values: "10, 20, 30"

Probabilities: "0.2, 0.5, 0.3"

Output: "Expected Value = 22"

Remember, the entered probabilities must be between 0 and 1 since a probability of 0 means that the event never happened, while one means it is certain.

The sum of all probabilities must equal exactly 1. The expected value calculator will show a warning if the sum is incorrect, which will disappear once the values are accurate. You may get an error message if the sum is not equal to 1. You can hit the cross button and correct the value & try again.

The expected value will be displayed at the bottom when the probabilities are correct and add up to 1.

Expected Value Formula

Mathematically, the expected value of a random variable X is the sum of each possible value x of X, multiplied by the probability of that value, P(x).

Here, P(x_i) is the probability of value x_i occurring (i = 1, ..., n) and n is the number of all possible values the random variable assumes.

Below is the rewritten formula using the summation sign:

How to Use the Expected Value Calculator

Follow these simple steps to calculate the expected value:

1. Enter Values: Enter your values into the "Value" fields. New rows will automatically start appearing as you fill the last field. (You can enter up to 20 values)
2. Enter Probabilities: Enter the corresponding probabilities into the "Probability" fields. Ensure each probability is in decimal form (e.g., 0.2 for 20%) and sum up to 1.
3. Check Probabilities: Ensure the sum of all entered probabilities equals 1. A warning message will be displayed if the sum is not 1 and until the values are corrected.
4. View Result: Once all values and probabilities are correctly entered, click the "Calculate" button to view the expected value.

How to Find Expected Value: Examples

Coin Toss

Suppose a fair coin toss has two possible outcomes: Heads (value = 1) and Tails (value = 0).

The probabilities are both 0.5. (i.e., 0.5 for 50%), The expected value is calculated as follows:

E(x) = 1 * 0.5 + 0 * 0.5 = 0.5

The expected value is 0.5, indicating that, on average, you get 0.5 points per coin toss.

Lottery Ticket

Imagine you buy a lottery ticket with a chance of winning $1000 (probability = 0.001) or winning nothing (probability = 0.999). The expected value is calculated as follows:

E(x) = 1000 * 0.001 + 0 * 0.999 = 1

The expected value of the lottery ticket is $1, meaning that, on average, you can expect to win $1 per ticket.

Frequently Asked Questions

What is expected value?

Expected value is a concept in probability and statistics that calculates the average outcome of a random event based on its probabilities and values.

How does the Expected Value Calculator work?

The calculator multiplies each value by its corresponding probability and sums the results to find the expected value.

Can the expected value be negative?

Yes, the expected value can be negative. For example, imagine you are playing a lottery game where you either win $100 or lose $150. The probability of winning is 0.4, and the probability of losing is 0.6. The expected value would be calculated as follows:

0.4 * 100 + 0.6 * (-150) = 40 - 90 = -50

In this scenario, the expected value is -$50, indicating an average loss of $50 per game.