What is Expected Value in Statistics - I

Again, let’s go back to the three-step process that we followed to find whether the upGrad red ball game was profitable for the players or for the house:

1. Find all the possible combinations.

2. Find the probability of each combination.

3. Use the probabilities to estimate the profit/loss per player.

Now that we have completed steps 1 and 2, let’s move on to step 3, where we will use the probabilities that we calculated to estimate the profit/loss per player.

So, the expected value for a variable X is the value of X that we would “expect” to get after performing the experiment an infinite number of times. It is also called the expectation, average or mean value. Mathematically speaking, for a random variable X that can take the values $x_1,x_2,x_3,...........,x_n$, the expected value (EV) is given by:

$EV\left(X\right)=x_1\ast P(X=x_1)+x_2\ast P(X=x_2)+x_3\ast P(X=x_3)+...........+x_n\ast P(X=x_n)$

As you may recall, for our red ball game, the expected value came out to be 2.385. What does this mean? How does this help us with our original question, which was: How much money on average are the players expected to make?

Let’s explore this in the following video:

The expected value should be interpreted as the average value you get after the experiment has been conducted an infinite number of times. For example, the expected value for the number of red balls is 2.385. This means that if we conduct the experiment (play the game) infinite times, the average number of red balls per game would end up being 2.385. You can try it out in this interactive app.

As you can clearly see, after a large number of simulations, the average value does, in fact, converge towards the expected value, which is 2.385.