Expected Value (EV) is a powerful decision-making tool used in various fields, from finance to game theory to probability. It helps decision-makers understand potential outcomes of a scenario and quantify the risks involved while making choices. In this article, we will explore the concept of expected value and explain how to calculate it step by step.

What is Expected Value?

Expected value is the weighted average of all possible outcomes of a random event, where each outcome’s probability serves as its weight. In practical terms, it represents the long-run average result of a series of events or trials. Simply put, EV helps us determine what we can expect on average over time if we were to repeat an action or decision multiple times.

Calculating Expected Value

To calculate the expected value of an event, follow these steps:

Step 1: Identify all possible outcomes

List all possible outcomes associated with the event or decision being analyzed. If you are dealing with a game involving dice, for instance, this would involve listing down the number on each face.

Step 2: Assign probabilities to each outcome

Determine the probability of each possible outcome occurring. Probabilities must be expressed in decimal form and add up to 1.

Step 3: Multiply outcomes by their respective probabilities

For each outcome, multiply its numerical value by the corresponding probability.

Step 4: Sum up the products calculated in step 3

Add up all the products obtained in step 3 to get the expected value.

Example

Let’s say we have a simple game involving a fair, six-sided die. If you roll an even number (2, 4 or 6), you win $10; otherwise (1, 3 or 5), you lose $5. To calculate the EV of this game, follow these steps:

1. Identify all possible outcomes: {win $10, lose $5}

2. Assign probabilities to each: P(win) = 3/6 = 0.5, P(lose) = 3/6 = 0.5

3. Multiply outcomes by their respective probabilities: {$10 * 0.5, -$5 * 0.5} = {$5, -$2.50}

4. Sum up the products: $5 + (-$2.50) = $2.50

So, the Expected Value of playing this game is $2.50, which means that on average, you would expect to win $2.50 per game if you played this game multiple times.

Conclusion

Expected value is a valuable tool for decision-makers to understand and quantify risks related to various scenarios or decisions. By determining the expected value of possible events, one can make informed decisions about whether to proceed with a particular course of action or choose among multiple options based on potential rewards and risks involved.