Summary: Basics of Probability

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:

EV (X) = x_{1}*P(X=x_{1})+x_{2}*P(X=x_{2})+....................+x_{n}*P(X=x_{n})

Another way of writing this is as follows:

EV(X) = \sum_{i=1}^{i=n}x_{i}*P(X=x_{i})

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.

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