In the previous chapter we saw what probability is and why do we need it? In this chapter we will cover various topics on the basis of probability.

In this chapter we will be covering random variables, what is a probability distribution and what is the expected value? So, before understanding these topics let's take an example, since we are talking about probability, let's take an example of a casino, in the casinos it always happens that… you must have seen, in the casino the house always wins, which means the house is always a winner.

How does he always win? How do the players draw less money? This is in particular very good and a lovely example to understand the probability.

To understand how the casinos operate, we will play one game in which we will learn various concepts term by term.

So, suppose, our game is this way.

I have one bag and in that bag I have five different balls.

Three balls are in red colour and two balls are in black colour.

Suppose we play this game with 30 people.

What is the job of these 30 people? First of all they will take out one ball from the bag.

They will note its colour and keep it back in the bag.

They will do this work four times.

Which means, suppose I am playing this game.

So, I took out the ball from the bag for the first time.

I noted its colour, it could be red or black.

I noticed it and put it back.

Second time again I pulled it out, noted its colour and put it back.

So, I will repeat this thing four times.

Now how will the money majorly come from this game? Suppose one person drew red balls all four times, which means the balls that I pulled out, they are red all four times.

If this is the scenario, or if this is the outcome then I will get 150 rupees in cash.

But suppose if I pulled out the ball the first time, it turned out to be red, the second time I pulled the ball out even that time it was red.

And the third time one black ball was pulled out.

What happened in that particular scenario? On drawing any different ball, I will have to pay a 10 rupees fine.

So, my game goes this way.

If I get all four balls of the same colour, of red colour, then I will get 150 rupees and even if one different ball is drawn, I will have to pay a penalty of 10 rupees.

So, what is in this particular scenario? I have to find out how much more money I can earn.

Either the organiser will be in a loss or he will be in profit.

Why are we seeing this example, so that different topics or concepts of probability can be understood easily.

Now, to solve this particular problem, the approach that we use would be a three-step approach.

What will be the first one? First of all we will find all the possible combinations which means in which way I drew the ball, which was the ball that I drew first and which was the ball that I drew next.

Based on it we will make all the combinations.

After that, how many combinations I have, we will calculate its probabilities.

After calculating its probabilities in the third step, we will use its probability to see how much money did every player earn or how much money did he lose? So, suppose this is my particular scenario where I am seeing which all can be the possible outcomes.

Now what can be my first possible outcome? It can be that all four balls are black, not even one red ball is pulled out.

Second scenario can be that there are three black balls and one red ball.

So, this can be in four different ways.

First red ball, the rest black balls.

First black ball, the second is my red ball and the other two would be black.

So, in this way there are four possible ways.

Third situation can be that where I can have 2 black and 2 red balls.

Which means you have pulled out the 1st ball which is red, you pulled out the second one, even that is red, the third ball pulled out was black, the fourth one was black.

I have a total of six cases even of this particular scenario.

Like you take one scenario, 1^{st} you drew black, second red, third black, fourth

red.

In the same way these would be my 6 cases in total.

My fourth scenario would be where I get one black ball and three red balls.

Which means you drew the 1st ball, it was black, rest of the 3 you drew continuously were red.

Even in this there can be 4 scenarios.

Suppose 1^{st}, 2^{nd} and 3^{rd}, all

three balls are red but the fourth ball is red.

My final scenario, which can be a possible outcome that can be that even one black ball is not drawn.

You will get all four red balls which means that you are in a winning situation.

So, all these would be my possible outcomes.

So, these possible outcomes would be a difficult situation for me.

Why? Because you're not able to quantify them in this way.

You're not able to give them any number.

You have gotten the different outcomes but before doing any such kind of statistical analysis it is always advisable.

These numbers or outcomes are converted into any such value which can easily give the result.

Basically, what we will do is we will assign one random variable x in these particular outcomes.

So, from here we will get our first concept of what is a random variable.

What does my random variable basically do? It converts any experiment’s outcome into a measurable quantity or in the form of one number.

Like I will take this example.

Suppose I have taken a random variable that is how many would be my number of red balls.

So, suppose this is the thing where my first ball is black and the rest of the balls are red.

So, what will be the number of red balls, three.

So, what will be my x? It will be 3.

Let's take one more case.

Where my first ball is read and in the rest of the two outcomes I have black ball, in the fourth outcome I have red ball.

So, how many are the total number of red balls? Two.

So, in this particular scenario, we will define x in a way where I can easily tell how we can define any of my outcomes in the form of numbers.

Like we were seeing in the previous example.

If I want to convert the total possible outcomes into variable form, how would the value be? First scenario is that my x would be 0.

Why? Because there is not even one red ball.

Second case is that I can see one red ball, so my x’s value is one.

In the third case, x value is two, since there are two red balls.

In the third case, x’s value is three since there are three red balls.

And in the final case, x’s value is four, where there are four red balls.

So, totally I have used my random variables and then I have converted the outcomes into numerical forms and I got different values for x.

Till now we have defined the random variables.

But with this I am not able to know how much money I'm going to win or how much money any participant or the person is going to lose? So, let's do one thing.

Whichever person is playing this game, based on their responses, we will draw one graph, we will take one value, let's suppose there is one person named Prakhar, when he drew the balls, he got three red ones.

This means my x value is three and I created one mark on it.

Megha, suppose she is a girl who drew zero red balls.

So, my x value is zero, there one value is increased, in the same way we noted different values of all 30 participants and we drew one chart which we will call a frequency distribution graph.

Basically, what happened was on different values of my x, how many times it has been repeated, number of times x has been repeated in the game.

I have plotted that in my frequency distribution curve.

Suppose 0 is repeated twice, 1 is repeated six times, 2 is repeated 7 times, 3 is repeated10 times, 4 comes five times.

So, the histogram that I drew was frequency versus x graph.

X is on the x axis, which is defining the number of red balls.

It keeps the values 0, 1, 2, 3, 4.

Frequently is on the Y axis, which is defining the height of different frequencies.

Like 0 has come twice, so its height is defining that this has come twice.

3 has come 10 times so its height is the highest.

Now with this particular graph, I got to know how many times the values have been repeated in my event.

If I wish to go calculate these frequencies on the basis of the probability.

So, we know what the formula of the probability is.

Suppose my X is 3 and I have to calculate its probability.

So, this means that frequency of 3, means how many times 3 is repeated upon total frequency.

This will be called my probability of x=3.

How many times has 3 been repeated? We had seen in the previous graph that it has been repeated 10 times.

What is the total frequency? 30, since 30 participants are playing in this game.

So, what will be my probability of x=3? 0.33.

In the same way, if I calculate the values of other random variables, I will calculate the probabilities, I will get different values like x of 0’s probability is 0.067, 1’s probability is 0.2, 2’s probability is 0.233, 3’s probability 0.333 and 4’s probability is 0.167.

I have drawn all these values in a graph.

The way we had seen frequency distribution’s graph.

In the same way, this is my probability distribution graph.

On the x axis there are number of random variables.

I'm calling it x and on the y axis I have got my probabilities.

And this curve is giving me different values of probability distribution.

So, with this I'm getting my second concept which is probability distribution.

What is it? This is a kind of a visual representation in which we define all the probabilities of x which means the random variable that I had chosen.

If I draw all probabilities of it that will be called my probability distribution.

There are no negative values in it.

This is the important point.

And the sum of all the probabilities is one.

Probability distribution can be in a table form like in the left side.

It can also be in the form like on the right side it can also be in the form of an equation.

So, probability distribution is a very important visual representation with which we can come to know many frequencies or probabilities are in my data.

Now, you will see one thing on the left, I've got my frequency distribution graph and, on the right, I have a probability distribution graph.

Both the graphs are completely similar except that on the y axis I have frequency in frequency distribution and on the y axis, there is probability in probability distribution case.

These two graphs are similar in shape but their scales are different.

My frequency is on the scale of 10.

My probability is on a smaller scale.

So, in this way, we have covered two things.

First, what is a random variable and how can we calculate its probability.

Now, we will come on to the third concept which is expected value, to understand this, we will do one thing.

So, now suppose 1000 people are playing, and someone asked me what would be the probability of drawing three red and two blank balls or how many people are expected to draw that value.

Now, here we already know the probability of getting 3 red balls and 2 black balls.

If I multiply that value with the total number of players then I will easily know how many would be the numbers of players who will draw three red balls and two black balls together.

So, suppose let's take the same particular scenario where we know the probability x=0 which is 0.067.

How many are my total number of players, 1 thousand.

So, the number of players who will not draw even one red ball is 67.

Why? Because we have multiplied the probability with the number of players.

If I see the probability of my x=1, which means only one red ball has to be drawn.

In that scenario, the number of players is 1000.

I multiplied those two probabilities and numbers.

So, there will be 200 such people from 1000 Who will draw one red ball.

The people who will draw two red balls will be 233 people.

The people who will draw three red balls will be 333.

The people who will draw four red balls will be 167 people.

I have calculated the total number of people, okay.

Now you tell me how many red balls are expected on an average.

Okay.

Here the term is expected, we call this particular scenario as expected value.

Lets understand it with this particular concept.

What is my total number of red balls in this scenario? Whatever was my value, means if even one ball is not coming out, then it is zero.

But how many people are there to draw it, 67.

Plus the people who will draw one ball is 200 people, the people who will draw two balls are 233 people.

There are 333 people who will draw three balls.

The people who will draw 4 balls are 167 people.

So, its total would be 2333, which means the total number of red balls which will be drawn in this particular event or game would be 2333 balls.

Now what would be the average number of red balls which will be drawn in every game, how much would be that? My total value that has come, the number of red balls divided by the total number of players.

That particular value would be my average number of red balls.

This means that if anyone plays this game, he will draw 2.33 balls on an average.

You must be thinking, how can 2.33 red balls be drawn.

Number of balls can either be 2 or 3.

So, generally what happens is that we calculate the expected value for infinite times of event.

Which means if I have played one event and I believe that it will go on for an infinite number of times.

So, in that my expected value that comes from x, which means my random variable value that comes we call that as expected value.

Basically, it is also called as mean, expectation or average.

If I call it in general mathematical term, so suppose I have a random variable x, which has different values from x1, x2, x3, up to xn.

Like in our particular scenario x=0.

Not getting any red ball’s random variable was x=1.

Getting 1, random variable was...

In the same way for different values my expected value would be x1 multiplied by probability of x1 plus x2 multiplied by probability of x2 plus x3 multiplied by probability of x3 up to xn.

Which means after multiplying all the values with its probability, my value that will come that would be called as my expected value and it helps me that find the average of a particular even.

Which means how much can I expect in that event.

So, this covers all the three topics random variable, probability distribution and expected value but our basic problem or question is still unsolved.

Which is how much money will I or how much money I can lose? So, what do we do for that? I think that our choice of random variable, we had counted how many red balls will be drawn, instead of it if we choose some other random variable.

Suppose, I have chosen the random variable in this situation.

The money that I have won after playing a game.

Which means my x’s value this time is how much money you have won playing the game once.

So, what will be my x’s value? +150 and -10.

Why? Because I get 150 if I win and if there is any such value where you lose than there is a penalty of -10.

I this scenario what will my expected value tell me? If it is a positive expected value that means that the player will win the money and the organiser will lose the money.

But if I get negative expected value in the result, this will mean that the player will lose his money and the organiser will win the money.

We already know that when will be the probability of x=150? When I will get all 4 red balls.

So, we had already calculated its probability which was 0.167.

Now the probability that my x should be -10, means, when will it be -10? When you have 0 red balls, 1, 2 or 3.

In any scenario, you have to give 10 rupees.

How will we calculate this probability? By adding all four probabilities, whatever value I will get, my probability will be x=-10, which comes to me as 0.833.

In this scenario we have calculated the probabilities of both the values.

I had two random variables x=150 and x=-10.

In the second step I calculated its probability.

When x = 150, then my probability comes to 0.167, when my x is -10 then my probability comes to 0.833.

In the third step I will calculate its expected value.

Because of which we will get to know whether we are going to win the money or lose it.

What would be the expected value? Multiplying probability value with random variable.

It means 150 multiplied by probability of 150+ -10 multiplied by probability of -10.

When I calculated it, my final expected value that came was 16.72.

This means that if any player plays this game, then on an average there is a change of the player winning 16.72 rupees.

Which is a great thing because if you play this game, you can win 16 rupees.

So, in this way we use probability in different scenarios and learn different concepts.

In the next chapter we will cover discrete probability distribution.

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