So, now let's discuss our probability concepts. Let's continue that. So, first of all, let me write again what were my axioms of probability. Just so that we don't forget. So, axioms again means the assumptions of probability.
So, what was my first axiom of probability? axiom one it was for any event event what is event what is an event tell me what is the definition of an event quickly give me uh in chat what is the definition of an event So for any event probability of that event is greater than equal to zero. This is my first axiom.
My second axiom is that probability of the sample space is equals to one. Also tell me what is sample space. Okay.
And what is my third axiom? I have infinite events which are which have nothing in common. So if I have different infinite events whose intersection is null. So if a1, a2 up to an and it is continuing further also if this is an infinite collection of disjoint events. Disjoint events means events whose intersection is null.
Means there is nothing in intersection. Disjoint events. Then the probability of A1 intersection. Sorry, A1 union.
A1 union, A2 union. And it goes on. This is nothing but you just add those probabilities.
So, I equals to 1 to infinity. Simply add. the individual probabilities.
Okay, these are the three axioms. Come on guys, you haven't told me what is event and what is happens face. So what is an experiment? Ma'am, I mean axiom one probability of any event is you have written that it is greater than or equal to zero but it should also be less than or equal to one also.
Yes, so what will happen we will prove that. So, we were saying that the minimum things we assume in axioms, we take the minimum things that we have to assume. The rest, what you said, p should be less than or equal to 1, we can automatically prove using them. So, we don't assume what we can prove.
So, if there is something that you can prove, don't assume that. So, we assume that. use all these axioms to prove. So very nice.
We will prove that also today. Okay. Also, can you please explain axiom 3 using an example?
Okay. So, suppose this is my sample space. What is sample space?
Set of all the possible outcomes. And what is event? Any subset of the sample space. Okay.
So what is happening is that there are many events that are disjoint. Disjoined means there is nothing in the intersection between them. So, let's say there is one event, this area is A1, this is A2, this is A3, this is A4. This series is going on like this.
Okay. Means there are small events like this. And there is nothing between them.
Otherwise, there is something common in the intersection. There is nothing like that. So, if that is the case, when you have infinite such, you know, different events happening. So, if you... probability, you have to find the probability of their union.
So what do you do? You just simply add their probability. So probability of a1, a2, a3, a4 up to a infinity, that is nothing but you just add pa1 plus pa2 plus pa3.
So let's assume that probability, these events, a1 could be that suppose you have an experiment, you are throwing a die. And event A1 could be that when you die throw, 1 will come. Okay?
So, the probability of 1 coming. What is event A2? that when you throw a die, 2 comes.
What is a3? 3 comes. What is a4? 4 comes.
Similarly, what is a7? When you throw a die, 7 comes. In this way, infinite events can be made because your numbers will never end. So, if you want to find out the probability of all these unions, then what do you do? You simply add these probabilities.
Probability of a1 plus probability of a2 plus probability of a3. This is what the third axiom says. In that case, it will be only one. So, why would we add? We know that the probability is one.
No, it is not necessary. You can assume that your A1 event instead of calling it one, I can say that the first event of A1 A1 event is if you throw a die, you get 6. Second event is if you throw a die, you get 7. 8, 9, 10. Do it like this. Then you will get probability 1. Yes. Right?
So we have to keep that also in mind. Right? Okay.
So this is my axiom. It says that if there is an infinite collection of disjoint sets. If we want to find the probability of their union, that is nothing but just add their probabilities. Okay? These are the three axioms that we have.
Now, what we will do is, these three axioms, these are the most basic assumptions. By using these assumptions, what all can we prove? We will do that. So, first of all, I want to see. So, first thing that I want to prove is, so, proposition.
Proposition means a statement that you are given. So proposition is nothing but a statement which can be true or false. So we will try to prove this proposition.
And what is my proposition? My proposition is that probability of a null event is zero. Now we can also say that why haven't we assumed that probability of null event is zero. Right? Yes ma'am.
So now what we will do is, using whatever 2-3 things that we know, we will come up with a proof for this. So we didn't have to believe this, we were able to find out using these 3 assumptions. Okay, so what we will do first is, so we are going to prove this basically.
My screen has failed I think. Yeah, okay. It's working.
Alright. So, I want to prove this that the probability of null event is 0. Okay? So, first I want to say this is null event. Okay? Sometimes these things can be asked by you.
So, you should just understand this. Sometimes people ask you that means in different ways sometimes proofs can be asked. and these are directly given in your book they might ask you this so null event this is null event so now I have to prove that probability of null event is 0 so see what I did I know my axiom 3rd what does axiom 3rd say that if I have infinite disjoint set disjoint events if I have infinite disjoint events and if I want to find their probability of union that is nothing but going to be I add them and basically I add their probabilities right this is true right so like that is the assumption so what I have assumed that I have a lot of events I have event First is 5, event second is 5, event third is 5. So, as an example, we can take this as my first event is that when I throw a die, 7 comes.
There is no member in it, right? Because 7 will never come. So, 7, 8, 9, 10, I have events like this. So, what I did?
My events are going on like this and in every event, Basically, we have five elements, no elements. Okay. And you tell me this, when you are doing the intersection of five with five, what will come? If you do the intersection of nothing with nothing, then what will happen? Term 0. So that will also be nothing, right?
That will also not have any member, right? So that will also be 5. Yes. Because in the terms of set, 0 means nothing, right? And that means it is 5. So basically, I have shown that this event A1 and A2 are disjoints. Because I have to show that these are disjoints in order to use axiom 3. Because what was my axiom 3?
That all these events are infinite collection of disjoint sets. Disjoint events. So just by doing this, I have said that all these events are disjoints. Ma'am why did we take intersection and not union?
Because property of disjoint, when can you say that events are disjoint? When their intersection is 5. Okay ma'am. Right, so all the events are disjoint. Okay, so basically what is my aim now, I have to find out.
What is the probability of A1 union A2? Up to is going on like this. And this is going to be nothing but using the third axiom.
If I want to get this property, like what, if I want to get this probability, using the third axiom, I can just say that this is nothing but I equals to 1 to infinity P of AI. Yes or no? Is everyone clear so far? Okay, Shaurya, are you following? Shubham, are you also following?
Okay, alright. So, okay. And what is this?
So, AI is nothing but, any event that I am watching, that is nothing but a phi event. So, this is nothing but probability of phi, right? I am just adding all the phi probabilities. Right? Yes or no?
I can write any AI event as phi. Okay. Okay?
And when I am looking here, then what is the situation? I can see that I have to find the probability of phi union, phi union, phi union. It is going on like this.
And what is this? Nothing union, nothing union, nothing union, nothing is ultimately going to give me nothing only. Yes or no?
Yes? Okay. So, this whole union will basically give me a null set. So, I have probability of phi.
So, probability of phi is equals to summation of i equals to 1 to infinity probability of phi. Okay. So, basically, what is happening is this probability of phi is equal to sum of infinite p-phis. So, this one value and these infinite values, when you sum them, what is the only possibility that you have in which case these two can be zero like these two can be equal what is the possibility in which these two can be equal otherwise there is no possibility and knowing that probability of any event has to be greater than or equal to zero if you put any non-zero value of p5 then what will happen? suppose we put value of 0.0001 then what will happen?
0.001 plus 0.001. How many times is this happening? Infinite times.
So it's like this is 0.001 and this is infinity. So these two things are not equal, right? The only case, the one and only case wherein this condition can hold is when p5 is 0, right? Do you guys agree with me? to take any other value of p5 greater than 0 these two equation can never hold like this equality can never hold yes or no okay so by this argument i can say that p5 has to be 0 there is no other option right this is your proof for you P5 being 0. Did you guys get it?
Ma'am, but P5 is 0, right? Its value is 0.001 or something else. No, no. So basically, we want to prove that P5 is 0. Here, we have not written that P5 is 0. But we are saying that can there be any value other than P5?
This is what we have to derive. So, p5 value is such that this equation holds. So, this equation is not holding for any other value except for p5 equals to 0. Hence, p5 has to be 0 in order to maintain this equality.
Okay ma'am. Right. So, we are not going because it's 0, hence this equation is true. We are saying this equation has to be 0. If there was any other value that would satisfy it, then we would have taken that value. But there is no value that is satisfying it except for P5 being equal to 0. So using that logic, we are saying that P5 has to be 0 otherwise this equation will not hold.
And this equation should hold according to my third axiom. It should hold according to the third axiom. Okay.
This is the argument. Okay guys, so is this clear? I can repeat if there are any doubts. Okay, lovely.
Okay. So, see, now this third axiom, what is it saying? A1, A2, AN, all these numbers are for infinite collection.
Infinite collection means there will be infinite events. Then you can use this axiom. But now what we want to prove is that this holds for a finite collection of events.
So what is the other thing that I want to prove? I want to say that suppose I have events like a1, a2 up to ak. So, first my events were going till infinite.
Now, I have limited events. Let's say I have k events. And they are disjoint. And so now basically I want to prove that the probability of A1 union A2 union a k is nothing but sum i equals to 1 to k p a i so this thing i want to prove so i want to prove this what do i want to prove that the axiom 3 which was holding for infinite infinite collection of events now i want to prove that all these events if I have finite events that means limited events and they are disjoint then I want to say that I can write it like this that suppose there are 2-4 events which are disjoint so probability of their union should be equal to sum of their probabilities do you guys understand the statement that I want to say this yes or no that if there are 2-3 events they are disjoint and I have to find the probability of union of all those events that should be equal to sum of their individual probabilities this is what I want to show and prove so when we were looking at axiom 3, we have not assumed anything like this but using axiom 3 and what we have just proved that T5 has to be 0 we want to show that this holds true you can prove this So, we usually use this, but this is actually something which comes out of the ICOS. So, the statement is clear to everyone.
What is the statement saying? If it is proof, we will go later. Okay, statement is clear.
Alright. Now I know what axiom 3 says. Axiom 3 says that if you have infinite collection of events, they are disjoint, then probability of their union is the sum of their probability, sum of that infinite collection. Okay?
So what did I do now? I have A1 to A2 events. Then I prepared some events artificially. I made some events artificially such that there is no member in them.
They are all 5 events. For example, if it was a dice, 7, 8, 9 are coming in the dice. So I artificially appended all these events.
append means you have added so I appended events I appended events that are disjoint disjoint and which events I have added so suppose my 1k plus 1 event what happened in which 5 events 5 members a k plus 2 which has 5 numbers like this I am adding infinitely so as soon as I added them this entire thing became an infinite collection of events yes or no so I am artificially creating all this in order to prove the above thing so this part is clear so I have appended events that are disjoint and they are individually null events. Yes or no? Yes. Okay.
So, what did we do after that? What did we do? Okay. So, now I know that all these events are disjoints.
Okay. And this is a phi event, so it will be disjoint from all of them. Okay.
So, now what is it? I can use the third axiom. and according to third axiom, what can I say?
that probability of A1 union, A2 up to Ak and after that, with whom am I taking union? I am taking union with this event, this event, this event there is no member in them so after doing all union, ultimately I will have only this union yes or no? There is no need to add anything in the union because they are all 5 events.
Yes or no? Right? Okay. Then? So, using the third axiom, I can say this is nothing but summation i equals to 1 to infinity probability of ai.
Okay? And I can divide this into two parts. which is, I sum i equals to 1 to k, probability of a, probability of ai, and the rest is, probability of ai, summation i is going from k plus 1 to infinity, then I can see, which are all these events, there are 5 events, and I just proved that probability of i is 0, then I can see, here what will happen?
0 plus 0 plus 0 plus 0. So this entire thing is going to be 0. Right. So this entire thing is going to be 0. And because it is 0, I can see that I can see that this part of mine has vanished. So this probability is equal to this particular probability. And that is what I had to prove. Okay.
That probability of A1 union. a2 up to ak is summation i equals to 1 to k probability of ai. Is that clear?
We have proved this by applying this logic. Okay? So, artificially we have added such events so that this entire set becomes infinite set.
And this formula I can apply to infinite set according to third axiom. So, I converted it into this using that. Then I broke it into two parts. And then the probability of this part was 0 using T5 equals to 0. And that is why I have this particular relationship between a disjoint set of probabilities. If you have disjoint sets, when you take their union, then its probability is nothing but some of the probabilities, some of the individual probabilities.
Okay? Is this proof clear? How is it proved? Yes or no?
Yes. okay lovely okay guys so these are our main main major major proofs okay