🎲 Expected Value Calculator

Calculate Expected Value (EV)

The Expected Value Calculator helps you determine the long-term average outcome of a random process. Expected value is a fundamental concept in probability theory, finance, and game theory. It represents the average value you would expect to get from an experiment if you repeated it many times. Simply input the possible outcomes and their associated probabilities to find the EV.

Expected Value Calculator

Calculate the expected value (mean) of a probability distribution. Enter outcomes and their probabilities to find the expected value, variance, standard deviation, and visualize the probability distribution.

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Outcomes and Probabilities

Enter the possible outcomes and their corresponding probabilities. Probabilities must sum to 1 (or 100%).

Total Probability: 0.00

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About Expected Value

Expected value (also called expectation, mean, or first moment) is a fundamental concept in probability and statistics. It represents the average outcome you would expect if you repeated an experiment many times. The expected value is calculated by multiplying each possible outcome by its probability and summing all these products.

Formula

For a discrete random variable X with outcomes x₁, x₂, ..., xₙ and corresponding probabilities p₁, p₂, ..., pₙ:

E[X] = Σ xᵢ · P(xᵢ) = x₁·p₁ + x₂·p₂ + ... + xₙ·pₙ

Variance and Standard Deviation

Variance measures how spread out the values are from the expected value:

Var[X] = E[(X - E[X])²] = Σ (xᵢ - E[X])² · P(xᵢ)

Standard Deviation is the square root of variance and is in the same units as the original data:

σ = √Var[X]

Properties of Expected Value
  • Linearity: E[aX + b] = aE[X] + b, where a and b are constants
  • Additivity: E[X + Y] = E[X] + E[Y] for any random variables X and Y
  • Non-negativity: If X ≥ 0 always, then E[X] ≥ 0
  • Monotonicity: If X ≤ Y always, then E[X] ≤ E[Y]
Applications
  • Finance: Evaluating investment returns and portfolio performance
  • Insurance: Calculating premiums based on expected claims
  • Game Theory: Determining optimal strategies in games and competitions
  • Decision Making: Comparing alternatives under uncertainty
  • Quality Control: Predicting defect rates and production outcomes
  • Economics: Analyzing consumer behavior and market outcomes
  • Risk Assessment: Evaluating potential losses and gains
Interpretation Guidelines
  • Expected value is the long-run average - not necessarily a possible outcome
  • A positive expected value suggests a favorable situation on average
  • A negative expected value suggests an unfavorable situation on average
  • Higher variance means more uncertainty and risk in the outcomes
  • Standard deviation provides a measure of typical deviation from the mean
  • In games and betting, negative expected value means the house has an edge

Understanding the Expected Value Calculator

The Expected Value Calculator helps you determine the average or “expected” result of a situation involving chance. Whether you’re analyzing dice rolls, investment outcomes, or game probabilities, this tool provides a clear and accurate way to understand what you can expect over time. It simplifies probability concepts and makes them useful for everyday decisions in finance, gaming, risk assessment, and statistics.

The Formula Behind the Calculator

Expected Value (E[X])

E[X] = Σ (xᵢ × P(xᵢ))

Where:

  • xᵢ = each possible outcome
  • P(xᵢ) = probability of that outcome

Variance (Var[X])

Var[X] = Σ (xᵢ - E[X])² × P(xᵢ)

Standard Deviation (σ)

σ = √Var[X]

These formulas allow you to calculate the expected value (average), measure how spread out the outcomes are (variance), and understand the typical deviation from the mean (standard deviation).

How to Use the Calculator

This calculator offers multiple modes to suit different scenarios. Each mode adjusts automatically to your needs:

  • General Expected Value: Enter any outcomes with their probabilities. Perfect for custom probability sets.
  • Dice Roll: Choose the number of dice and sides to find the expected outcome for gaming or random rolls.
  • Coin Flip: Input the probability of heads or tails and their respective payouts to assess fair or biased coins.
  • Game/Betting Scenario: Enter your bet, possible winnings, and win probability to understand if a game offers a positive or negative expectation.
  • Investment Returns: Add different market scenarios, such as “Bull Market” or “Bear Market,” with return percentages and probabilities to estimate expected gains or losses.

Step-by-Step Guide

  • Step 1: Select the calculator mode that fits your situation (General, Dice, Coin, Game, or Investment).
  • Step 2: Enter each possible outcome along with its probability. Ensure the probabilities add up to 1 (or 100%).
  • Step 3: Click “Calculate Expected Value” to get results instantly.
  • Step 4: Review the results for Expected Value, Variance, Standard Deviation, and Median.
  • Step 5: Check the Probability Chart and Calculation Steps for a visual and detailed explanation.

How the Results Help You

Understanding expected value gives you insight into what you can reasonably anticipate over many trials or decisions. Here’s how it can help:

  • Financial Planning: Estimate long-term investment returns and assess potential risk.
  • Game Analysis: See if a game or bet is fair by comparing potential winnings and losses.
  • Decision Making: Choose options with the most favorable average outcome.
  • Risk Evaluation: Identify how much uncertainty or variability is present in outcomes.
  • Statistical Learning: Understand probability distributions and averages in practical examples.

Example Applications

  • Investments: Predict potential profit or loss based on market conditions and their probabilities.
  • Games of Chance: Calculate the fairness of dice or card games by measuring expected gains or losses.
  • Insurance: Estimate average payouts or claims to determine premium pricing.
  • Business Forecasting: Model possible scenarios to guide strategic decisions.

Frequently Asked Questions (FAQ)

1. What is the expected value?

Expected value is the long-term average of all possible outcomes, weighted by their probabilities. It shows what you can expect to happen on average if you repeat an event many times.

2. What does a positive or negative expected value mean?

  • Positive: You are expected to gain on average (favorable scenario).
  • Negative: You are expected to lose on average (unfavorable scenario).

3. Why do probabilities need to sum to 1?

Because all possible outcomes together must account for 100% of the possibilities. This ensures that every potential event is included in the calculation.

4. How is variance different from expected value?

Expected value shows the average outcome, while variance measures how spread out those outcomes are around that average. High variance means more uncertainty and fluctuation in results.

5. Can I use this calculator for real-world decisions?

Yes. The calculator is useful for assessing investments, betting odds, and any decision involving uncertain outcomes. It helps compare risks and expected returns before taking action.

Conclusion

The Expected Value Calculator is a practical and insightful tool for anyone who deals with uncertainty — from students and analysts to investors and gamers. It helps transform probabilities into meaningful insights, empowering better, data-driven decisions.

More Information

The Expected Value Formula:

The expected value, E(X), is calculated by multiplying each possible outcome by its probability and then summing all of those values.

E(X) = Σ [x * P(x)]

  • x: The value of a specific outcome.
  • P(x): The probability of that outcome occurring.
  • Σ: The summation symbol, meaning you add up the results for all possible outcomes.

The expected value is not necessarily the most likely outcome, but rather a weighted average of all possible outcomes.

Frequently Asked Questions

What does expected value tell you?
Expected value (EV) tells you the average outcome you can expect from a random variable over the long run. In gambling or investing, a positive EV suggests a profitable venture over time, while a negative EV suggests a loss.
Can the expected value be negative?
Yes. A negative expected value means that, on average, you would expect to lose that amount per trial over many repetitions. For example, most lottery tickets have a negative expected value.
What is the difference between expected value and average?
The average (or mean) is calculated from observed data that has already occurred. The expected value is a theoretical value calculated from a probability distribution, representing the average outcome you would expect to see in the future if the experiment were repeated many times.

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