Expectation

The expectation is an important concept in the probability theory and let’s try to motivate the concept through an example say gambling and we are interested in the question: “Does gambling pay off?”.

We have the roulette in casinos where the idea is that it contains a ball and it spins and when it stops, the ball lands on one of the numbers

And it has the numbers from 00 to 36

The one type of bet the person can place is on the exact number and the standard pay off in such bet is 35:1 that means if someone bets Rs. 1 on a number and if the ball lands on that number, then the person gets Rs. 35

The question of interest is “Does gambling pays off?”

So, if you’re to play roulette a large number of times, then eventually will you be winning money or losing money?

As almost everyone has heard that the odds are in the favor of the house(casino) meaning the casino wins in the long run. Let’s try to understand the basis of this statement.

To answer this question, we are interested in knowing the profit in the long run, and this profit we can think of as a Bernoulli’s trial, everytime I play the game, there are two outcomes: either I’ll succeed or I’ll fail i.e either the ball lands on the number I have chosen or it will not land on that number. And if I succeed, then my profit…