Expected Value Calculator
Enter the value of each possible outcome and its corresponding probability to calculate the expected value.
Calculated Expected Value:
Understanding Expected Value
Expected Value (EV) is a fundamental concept in probability theory and statistics, widely used in decision-making, risk assessment, and game theory. It represents the average outcome of a random variable over a large number of trials. In simpler terms, if you were to repeat an event many times, the expected value is the average result you would anticipate.
How is Expected Value Calculated?
The calculation of expected value is straightforward. For a discrete random variable, it is the sum of the products of each possible outcome and its corresponding probability. The formula is:
EV = Σ [Outcome × P(Outcome)]
Where:
EVis the Expected Value.Outcomeis a specific value that can occur.P(Outcome)is the probability of that specific outcome occurring.Σ(sigma) denotes the sum of all such products.
It's crucial that the sum of all probabilities for all possible outcomes equals 1 (or 100%). If the probabilities do not sum to 1, the calculation will not accurately reflect the true expected value of the complete set of possibilities.
Why is Expected Value Important?
Expected value is a powerful tool for making informed decisions, especially when faced with uncertainty. It helps individuals and organizations:
- Assess Risk: By calculating the EV of different scenarios, one can understand the potential average gain or loss.
- Strategic Planning: Businesses use EV to evaluate potential investments, marketing campaigns, or project outcomes.
- Gambling and Games: EV is used to determine if a bet or game is favorable in the long run. A positive EV suggests a long-term gain, while a negative EV suggests a long-term loss.
- Insurance: Insurance companies use EV to set premiums, balancing the probability of claims against the payouts.
Examples of Expected Value in Action
Let's consider a few practical examples:
Example 1: A Simple Game of Chance
Imagine a 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), nothing happens (you win/lose $0). What is the expected value of playing this game?
- Outcome 1: Win $10 (Probability: 1/6 ≈ 0.1667)
- Outcome 2: Lose $5 (Probability: 1/6 ≈ 0.1667)
- Outcome 3: Win/Lose $0 (Probability: 4/6 ≈ 0.6667)
EV = ($10 × 1/6) + (-$5 × 1/6) + ($0 × 4/6)
EV = $1.6667 – $0.8333 + $0
EV ≈ $0.83
This means, on average, you can expect to win about $0.83 each time you play this game over many trials.
Example 2: Business Investment Decision
A company is considering investing in a new product. There's a 40% chance it will be highly successful, generating a profit of $500,000. There's a 35% chance it will be moderately successful, generating $100,000. There's a 25% chance it will fail, resulting in a loss of $200,000.
- Outcome 1: Profit $500,000 (Probability: 0.40)
- Outcome 2: Profit $100,000 (Probability: 0.35)
- Outcome 3: Loss -$200,000 (Probability: 0.25)
EV = ($500,000 × 0.40) + ($100,000 × 0.35) + (-$200,000 × 0.25)
EV = $200,000 + $35,000 – $50,000
EV = $185,000
The expected value of this investment is $185,000, suggesting it's a potentially profitable venture on average.
Using the Expected Value Calculator
Our calculator simplifies the process of finding the expected value. Simply input the numerical value for each possible outcome and its corresponding probability (as a decimal between 0 and 1). You can use up to four outcome-probability pairs. If you have fewer than four, leave the unused fields at their default values or set their probabilities to 0.
Remember to ensure your probabilities accurately reflect the likelihood of each outcome and ideally sum up to 1 for a complete analysis of all possibilities.