Expected Value

# HStatistics.com

### Expected Value

In this post we will learn about what is expected value and how it is calculated.

Before that, let’s learn something about calculating mean and averages.
If, let’s say, you scored 80 marks (out of 100) in English and 90 marks (out of 100) in Maths. Then what would be your average score?
You would say 85 marks, right?

Well, that is correct.
But we will delve into the method used to calculate it.
For example, if I modify the question and ask you what is the average (for the same marks mentioned above) if there are weights attached to both subjects ie. English has a weightage of 1 and Maths has a weightage of 2.
Try to calculate before reading on.

So, here we have added varying weights (1 to English and 2 to Maths).
The method used to calculate the average now would be:

(80 * 1 + 90 *2) / (1 + 2)

Did you notice what we are trying to do?

We have multiplied the weights with the respective values (marks) and divided the sum with the Total of the weights. This is called the weighted average.

Earlier - when you had calculated the simple average as 85 - this is the same methodology you had intuitively applied. There the weights were equal. Let’s say 1 for English and 1 for Maths.

Thus, we got, (80 * 1 + 90 *1) / (1 + 1) = 85.

This is what is called as weighted average.

So now, what is Expected Value?

Expected Value is nothing but the Weighted Average of the events being considered, where the Probability of those events happening are the Weights attached to them.

Let’s take an example to understand this better.

Say you are investing in a Project and the probability of generating 10 million profit is 0.3 (or 30%)
And the probability of losing 2 million is 0.7 (or 70%).
Should you invest in the project? What is the expected value here?

The expected value would be calculated as:

+10 * 0.3 + (-2) * 0.7 = +1.6 million

Thus, the expected out come of the project is that you would gain 1.6 million and hence could invest in it.

Let’s take another example.

Suppose you are playing a of dice.
You bet 60 dollars on the number 6.
Meaning, you will throw the dice once and if the number 6 comes on top, you will get 600 dollars in return. If any other number comes up, you lose your 60 dollars.

Should you play the game? What is the expected value?

First we calculate the probabilities of winning and losing respectively.
Probability of winning = 1/6 (since in only 1 outcome out of 6 you win) Probability of losing = 5/6

Now, the expected value would be:

+540 * 1/6 + (-60) * 5/6
(Note 540 comes after 600 - 60. Since that is the net profit you would make if you win)
= +40 dollars

What this means is that, if you play this game repeatedly (every time betting 60 dollars) infinite number of times, you would end up having a net positive gain of 40 dollars.
And, thus you should take risk and play the game.