In the first section, you learnt how to **quantify the outcomes** of events using **random variables**.

For example, recall that we quantified the balls of a particular colour that we would get after playing our game by assigning a value of X to each outcome. We did so by defining **X as the number of red balls** we would get after playing the game once.

Next, we found the **probability distribution**, which was a **distribution giving us the probability for all the possible values of X**.

We created this distribution in a **tabular form**

We also created it in a **bar chart form**.

You saw that in the bar chart form, we were able to visualise the probability in a much better way. Thus, this form is used more widely as it helps you see trends easily.

Then, we went on to find the **expected value** for X, the money won by a player after playing the game once. The expected value (EV) for X was calculated using the following formula:

Another way of writing this is as follows:

Calculating the answer this way, we found the expected value to be +11.28.

In other words, if we conduct the experiment (play the game) **infinite times**, the **average money** won by a player would be ₹11.28. Hence, we decided that we should either decrease the prize money or increase the penalty to make the expected value of X negative. A negative expected value would imply that on average, a player is expected to lose money and the house profits.