Expected Value Calculator

Calculate the expected value of random variables by entering outcomes and their associated probabilities.

Calculate Your Expected Value Calculator

Outcomes and Probabilities

Outcome ValueProbability

What is Expected Value?

Expected value (EV) is a fundamental concept in probability theory and statistics. It represents the long-term average outcome of a random variable. In simpler terms, if you were to repeat a random experiment many times, the expected value is the average of all outcomes, weighted by their probabilities.

How to Calculate Expected Value

The expected value is calculated by multiplying each possible outcome by its probability and then summing all these products:

E(X)=∑(xi×pi)E(X) = ∑(x_i × p_i)

Where:

  • E(X) is the expected value
  • x_i represents each possible outcome
  • p_i is the probability of that outcome occurring
  • ∑ means to sum over all possible outcomes

Example Calculation

Let's consider a simple example of a game where you:

  • Win $100 with probability 0.2 (20%)
  • Win $50 with probability 0.5 (50%)
  • Win $0 with probability 0.3 (30%)

The expected value would be:

E(X) = ($100 × 0.2) + ($50 × 0.5) + ($0 × 0.3) = $20 + $25 + $0 = $45

This means that if you played this game many times, you would win $45 on average per game.

Applications of Expected Value

Expected value is used in numerous fields:

  • Finance: Evaluating investment opportunities, calculating insurance premiums, and pricing options
  • Decision Making: Comparing different strategies based on their expected outcomes
  • Game Theory: Analyzing optimal strategies in games of chance
  • Statistics: Serving as the mean of a probability distribution
  • Machine Learning: Optimizing models based on expected loss or reward

Limitations of Expected Value

While expected value is a powerful tool, it has limitations. It only provides the average outcome over many trials and doesn't account for risk or variability. Two different scenarios might have the same expected value but very different risk profiles. Additionally, for some distributions, the expected value might be a value that can never actually occur (like getting 3.5 when rolling a six-sided die).

See Also

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Frequently Asked Questions

Expected value is the anticipated value of a random variable, calculated by multiplying each possible outcome by its probability and summing these products. It represents the long-term average result if you were to repeat an experiment many times.

Probabilities must sum to 1 because they represent all possible outcomes of an event. A total probability of 1 (or 100%) means that all possible scenarios have been accounted for in your calculation.

Yes, expected value can be negative. This often occurs in games or investments where losses are possible. A negative expected value means that, on average, you would lose money or value over many repetitions.

In gambling, expected value helps determine if a bet is favorable. Games with negative expected values (like most casino games) will result in losses over time. Professional gamblers look for positive expected value opportunities where the potential winnings outweigh the risks.

Expected value is a theoretical concept that predicts the long-term average outcome of a random process. An average is calculated from actual observed data. Given enough trials, the average of observed outcomes should approach the expected value.

In decision making, expected value helps compare different options objectively. The option with the highest expected value will, on average, provide the best outcome over many repetitions. However, other factors like risk tolerance should also be considered, especially for one-time decisions.

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