Comprehension: Expected Value

In the last session, you learnt about expected values. Now that you know how to calculate probability using the binomial distribution, it would be better to revisit the concept and practice some questions.

Calculating the expected value is a three-step process:

  • Define the random variable (X).
  • Calculate the probability distribution P(X). You’ll need to calculate it on your own.
  • Plug the above two terms in the following formula:


Let’s understand the process with an example. Suppose you’re playing a game involving a 6-sided die. Could you tell what is the average outcome that you’d expect each time the die is thrown? Answering this question requires us to calculate the expected value.

Let’s solve this problem step by step:

  1. The first step is defining the random variable. The random variable (X) is the outcome of a die throw. So, X = {1, 2, 3, 4, 5, 6}
  2. The second step is to calculate the probabilities related to each outcome. The probability of each outcome is 16 in a die throw

Now, you have X and P(X). If you plug these values in the formula E[X]=∑(X×P(X)), you’ll get 3.5 as the expected value. So how to interpret this number? This means if you were to throw the die a large number of times, the average of those numbers will tend towards 3.5.

So, why do we need the expected value at all? Well, the expected value lets you reason about real-world random phenomenon more rationally. For instance, investing in stock markets is a popular use-case where this concept is used.

Suppose you’re interested in investing in the stock market. It’s always better to invest in multiple stocks rather than one stock. You can calculate the expected value of your returns using the concept of expected value. Let’s take a simple hypothetical situation. Our random variable, in this case, can take the expected return of each stock. Then, to calculate the expected return, you need the probability of returns for each stock. This way, you can calculate the expected return of your entire portfolio, which will allow you to invest wisely.

Similarly, the expected value is often used in business settings where a stakeholder has multiple options.

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