Expected Value Calculator

The expected value calculator helps you to calculate expected value, which is the average prediction of a probability distribution.

What Is Expected Value?

The expected value is the average of the possible outcomes of a random variable or event ‘X’, weighted by their probabilities ‘P(X)’.

How to Find Expected Value?

Steps include:

1. Multiply the value by its weighted probability
2. Sum all the values to find the result

Mathematically;

\(\text{Expected Value Formula}=\sum P\left(X_{i}\right) . X_{i}\)

Worked Example - Calculating Expected Value for a Dice Game

Suppose you play a game that involves rolling a fair six-sided die. The payouts and probabilities for each outcome are as follows:

Calculations:

Step 01: Multiplying Each Outcome by Its Weighted Probability

\(EV = (-2 \times \frac{1}{6}) + (0 \times \frac{1}{6}) + (2 \times \frac{1}{6}) + (4 \times \frac{1}{6}) + (6 \times \frac{1}{6}) + (8 \times \frac{1}{6})\)

Step 2: Calculating Individual Products

\(EV = \left(-\frac{2}{6}\right) + \left(0\right) + \left(\frac{2}{6}\right) + \left(\frac{4}{6}\right) + \left(\frac{6}{6}\right) + \left(\frac{8}{6}\right)\)

Step 3: Taking the Sum of All the Values

\(EV = \frac{-2 + 0 + 2 + 4 + 6 + 8}{6} = \frac{18}{6} = 3\)

Practical Uses of Expected Value Calculator:

Our calculator serves different purposes under conditions where probabilistic decision-making is very important. Some of these include:

Investments & Stock Market:

• Financial experts can assess the expectation value of various bonds, stocks, or investment portfolios
• Investors can use the calculator to determine loss scenarios and profit potential

Business Decision-Making:

• Can be used to predict the expected probability of win for a project
• Companies can use the tool to assess the profitability in certain conditions

Insurance Premium Calculation:

• Companies can foresee the potential outcomes of different insurance premium plans and factor in the outcomes of payouts

Statistical Analysis:

• Expected value helps estimate parameters of data distributions, laying the groundwork for statistical methods like the Central Limit Theorem. 
• In regression analysis, the expected value helps determine unbiased estimators, ensuring the estimated value of a parameter is close to its true value. 

Medical Trials & Healthcare:

• Medical teams can use the tool to evaluate the effectiveness of different medical procedures and treatments
• It can also be used to determine the cost-effectiveness of new drugs or treatment options.

Related Questions:

Why is expected value important in probability and statistics?

Expected value calculation helps us to predict the expected average when a risk is taken with an uncertain outcome.

Can an expected value be negative?

Yes, it can be. If calculated expected number is negative, it means you are likely get an average loss, which is unfavorable.

Additional References:

DeGroot, M.H. and Schervish, M.J. (2012) Probability and Statistics. 4th edn. Boston, MA: Pearson Education. Available at: https://www.pearson.com/store/p/probability-and-statistics/P100000617944
 

Feller, W. (1968) An Introduction to Probability Theory and Its Applications. 3rd edn. New York: Wiley. Available at: https://www.wiley.com/en-us/An+Introduction+to+Probability+Theory+and+Its+Applications%2C+Volume+1%2C+3rd+Edition-p-9780471257080
 

Ross, S.M. (2014) Introduction to Probability Models. 11th edn. Amsterdam: Academic Press. Available at: https://www.elsevier.com/books/introduction-to-probability-models/ross/978-0-12-407948-9