Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random event. Mathematically, the expected value is the weighted average of all possible outcomes, where each outcome is assigned a certain probability. In this article, we will discuss step by step how to calculate the expected value for different scenarios.

Step 1: List all possible outcomes

Before calculating the expected value, you need to identify all possible outcomes of an event. These can range from numerical returns (e.g., profits or losses in investment) to categorical outcomes (e.g., success or failure of a project). It is crucial to list all potential outcomes without missing or duplicating any possibilities.

Step 2: Assign probabilities to each outcome

The next step involves assigning a probability to each outcome, which represents the likelihood of that outcome occurring. These probabilities are usually based on historical data, expert opinion, or estimation techniques. Make sure that all probabilities assigned are between 0 and 1 (inclusive) and add up to 1.

Step 3: Calculate the result for each outcome

In some cases, you might need to calculate the result for each possible outcome. This is common when dealing with complex calculations, such as investment gains or losses. Calculate the corresponding result for each listed possible outcome before moving on to the next step.

Step 4: Multiply each outcome by its probability

Now that you have a list of all possible outcomes with their respective probability and result (if applicable), multiply each outcome by its assigned probability. This gives you a set of weighted results which combine both probability and the impact of individual outcomes.

Step 5: Sum up the weighted results

The last step is to add up all the weighted results from Step 4. The sum represents the expected value of the random event under consideration.

Example: Expected Value Calculation for a Simple Dice Game

Let’s illustrate these steps with an example. John is betting on a simple dice game where he wins $2 if the dice rolls a 1, breaks even if it rolls a 2 or a 3, and loses $1 if the dice rolls 4, 5, or 6.

Step 1: Possible outcomes are winning $2 (W), breaking even (B), and losing $1 (L).

Step 2: Assigning probabilities: P(W) = 1/6, P(B) = 2/6, P(L) = 3/6.

Step 3: Calculation not needed in this case.

Step 4: Weighted results: ($2 × 1/6) = $0.33; ($0 × 2/6) = $0; (-$1 × 3/6) = -$0.50.

Step 5: Summing up the weighted results: $0.33 + $0 – $0.50 = -$0.17.

Therefore, the expected value of playing this game is -$0.17, which means John can expect to lose an average of $0.17 for each game he plays.

Conclusion

Calculating the expected value follows a simple five-step process: listing all possible outcomes, assigning probabilities to each outcome, calculating the result for each outcome (if necessary), multiplying each outcome’s probability by its result, and summing up the weighted outcomes. Mastering this concept will provide you with valuable insights into how to analyze risky scenarios and make better decisions in various aspects of life such as finance, project management, and gambling.