Introduction to Probability and Statistics: Key Concepts and Terminology

Hello students, my name is Dr. Gajendra Purohit and you are watching our YouTube channel. Today I am going to start a new topic that is probability and statistics where I will be discussing and covering everything in detail in my upcoming classes. And the old videos I uploaded, you can see the entire playlist here. There we started with random variables where we covered discrete and continuous random variables. We had also covered PDF, PMF.

Then in probability distribution, we discussed discrete and continuous distributions. And then we covered Joint PDF, Joint PMF, MGF and CGF. So, we discussed lot of things in our complete playlist. We also received love and appreciation from all the students. So I thought I should work on two point o in this and in this series I am starting from the basics.

Today, I will introduce some terminology used in probability. and we will talk about basic probability questions and types of events, then discuss addition, multiplication, conditional probability and Bayes theorem. then after that we will be discussing the topics like continuous and discrete random variables and then after that

the expectation and calculation of variance. Like this, there are different topics available. I will try. to complete it. So students let's start.

MUSIC So students, firstly here we will talk about the random experiment. Students, whenever we talk about the probability, the word random is very important.

If anything is being done with the planning, then the concept of probability is never applied. For example, you have an exam and before that you get the paper as the paper has been leaked. So when you will get the marks for it, you know you are going to get full marks, right?

and when you give the exam, and paper is not known to you beforehand, if you score eighty percent in that it means that out of hundred eighty percent of your total questions were correct then that shows the probability

When the paper itself is leaked, you'll get a hundred percent. Their concept of probability is not used and questions also come randomly. if by any chance The lottery system is discussed here.

And you know that this pink one here, it has the prize inside, then you will be picking that one only. Therefore, it would be a hundred percent probability of you winning the lottery. and if by any chance you are unaware of it like before and you are randomly picking a ticket and then if you win that prize let's say there are a hundred tickets and you picked one and you won then the probability of you winning that prize will be one by hundred right So, the random word in the probability.

it plays a big role like how we are throwing a dice here or how we are tossing a coin or Randomly picking a card from deck of cards or taking out a ball from a box full of different colored balls. whether it is of red or blue color, they are random experiment. So students these kinds of events are called as the random experiments.

But, whenever we are discussing about any of the experiment, it is necessary to have knowledge of sample space. and without proper knowledge of sample space we cannot determine that what will be the probability right. Let me give you an example. Suppose there is a lottery box which has many tickets kept inside and you have taken out randomly one ticket from it.

and you won. but what was the probability or what will be the probability of winning. This will only be known when you know how many tickets are there in that box.

If you find out that the total number of tickets are one hundred and out of those you can get one prize. So when you participate and pick one ticket your probability of winning will be one by hundred. For that, it is necessary to know the sample space that is the possible outcome. For example, when we toss a coin, then while doing that there are two possibilities, either head or tail, right?

So students, this head or tail that we will get, this is the sample space. And after that, when we will toss the coin here, whether this will be head or tail, this will be an event, right? What is sample space? Sample space is set of all possible outcomes of the experiment. Like I am explaining to you.

coin is tossed and we'll get head or tail if we will toss it two times. A coin is tossed once and we will get head or tail, right? If we toss it twice then its sample points are two to the power one. If I am tossing this two times, what will we get? Head, head.

head tail. tail head and then tail tail, right? Two power two is its sample points.

if you are told that a coin is being tossed three times. what will be its sample points. Sometimes students face difficulty in this, like I remember So it will be head, head, head.

head head tail. Head, tail, head. tail, head, head, then head, tail, tail, then tail, head, tail, then tail, tail, head, then tail, tail, tail, right? So students, this becomes our sample space.

but you'll think that I already know it all. Look students, we have two to the power three sample points. Students, if you are having difficulty in writing sample space, then don't worry. You know this two figure. Nothing much needs to be done. Pay attention to this too. You already know it.

Right? you know the two's method, multiply it with this one, right? This first one will be multiplied by this, then this will be multiplied by this, then this will be multiplied by this, this with this. We will keep multiplying like this and then the complete sample space will be obtained.

Students, I will give you another small example for it, that if I am throwing one dice, right? When dice is thrown, then what are the possible outcomes? It can be one or two. It can be three. It can be four. It can be five. It can be six. It means six power one will be its total outcome.

and if here, Two dice are thrown, right? Sometimes there is confusion in question.

that The two dice, they are thrown once. or

One die is thrown two times. the sample space will remain same. If a single coin is tossed twice,

or the two coins are tossed one time. sample points will be same. So don't get confused. If we will throw two dice together, then sample space we will get will be

How do we write that? There is a proper way to it. If you learn that method, then you will not have any difficulty in solving any question right. So, what will we do here? Students, here we have one comma one.

one comma two. one comma three, one comma four right. So, with one we will write one, one, two, one, three like this. one comma five, one comma six.

then this will be two comma one, two comma two. then, two comma three, two comma four, right? two comma five.

and two comma six then after that we will get three comma one. three comma two. Three comma three.

Three comma four. three comma five. right and three comma six.

then we will get four comma one. four comma two. four comma three.

Four comma four. four comma five. four comma, six, right?

then we will get five comma one. five comma two.

Five comma three. five comma four five comma five.

Five comma six. then we will get six comma one. six comma two.

Six comma three. six comma four. Six comma five. Six comma six

These will be the total sample points and these will be six to the power two. you might have a question. that if by chance we get this question that we throw dice two times.

What will be the probability if its sum is eleven or more? So students, when will the sum become eleven? Look. So the sum will be this, this or this. It's either eleven or twelve, right? we can easily calculate its probability if we know sample space

then we can easily calculate the probability of any event. And there are more questions especially about the sample space. that are card-based questions. Some students struggle in this concept

How many types of cards are there? Well, the students who play cards here will know it. There are total of fifty two cards, out of them the twenty six cards are black.

right? and the remaining twenty six cards they are the red cards. Right? Students keep this in mind please. And now if we discuss about the black cards here.

So let me tell you, within these black cards, again we have here the thirteen cards which are called as Spade. It is called spade. And the other thirteen cards that we have are of club, right? Club cards are called Chidi in Hindi.

It looks like this. Right? So students, if you have seen it ever, it looks something like this.

And here this. If you look here at this Kala paan, we call this spade. So students,

Here out of these twenty six cards, the red cards that we have Those red cards are again divided in thirteen cards each. So that are the red cards, right? What do we call them? Red cards, right?

We call them red cards, right? The red cards. And if you've seen, we also call this heart. We also call this heart. Right? We can call this heart.

Students, we call this one as red heart or lal paan in Hindi, right? And then over here, the thirteen cards we have, they are diamonds, right? We call them diamonds. Diamonds, right? You might have seen these before many times.

So like this, we have the thirteen cards of these types, right? Please pay attention. See, we have these cards here. So students, is that clear? Now please pay attention to one thing.

Here we have these cards. In these cards, understand that whenever we discuss about spade card, then Here. We have

this one, we call this as ace, right? After that from number two to number ten all of these are the cards and nothing else. After that next one is J.

Then, q and k right. So students, here we have Jack, Queen and King. That's what we call them.

It's Gulam Begum Badshah. In the same way, students, what do we have here? Here, one ace will be of club card. After that, from two up till ten, there will be only cards. After that again it will be j, q, k.

In same way, if we discuss about red cards, then in that we also have one ace, right? And after that from two up till ten we have normal cards. Then again we have J, Q, K. and students in the same way if we discuss about

then here also we will get same this is one ace and from two up till ten we have normal cards again here jqk And these JQK cards which are right here, we call them as face cards. If you are asked how many face cards are there out of fifty two cards, then these all are the face cards here.

If you count them, it's three plus three plus three plus three equals twelve. So these twelve cards that are here, they are face cards. If you are asked how many black face cards are there, so there are six black face cards. If you are asked how many red face cards are there, it's six.

If we add this here, please pay attention and we add this too. if we add this and also add this and also add this then the total number of cards we get will be sixteen right, four each. These are the owner cards.

This is called an owner card, right? Please note this carefully. Ace, Jack, Queen and King. These are called owner cards. So students will call them owner cards. So for this topic, you should have knowledge about the cards. It's important.

Let's move on. I have already explained about the event right students whenever we will toss something here right then at that time Please look here, I will show you. Let's say for example I have tossed a coin two times.

Students, I tossed a coin here. When coin was tossed, then we get head head, head tail. tail head and tail tail. That's what we get, right?

Now here, when two heads are appearing, it is called an event, right? getting heads once in two cases we will get one head right and this will be an event Students, you need to understand the event, whatever that we want.

that the heads appears twice or it appears once that will be an event. So is that clear students? Let's move ahead. The definition of event is given. We have a die and when we roll it, we get from one to six.

Students, if we have an even number of outcomes, then these three are the possibilities, two, four, six, and this is called an event. If odd outcome, then one, three, five will also be a possibility. this will be the outcome if number is greater than four so students whatever we want like if we are throwing a die two times and we want that the sum of its outcome is greater than eleven

then we will write as the sample, the total sample space that we want to be more than eleven. So whatever outcome we get, we will call it an event, right? Students events are also of two types. One is a simple event and other is the compound event and event having only one sample point if we only have one sample point.

then that type of event, what will that event be called? It is called simple event. For example, if a coin is tossed two times, the outcomes would be head-head, head-tail, tail head and tail tail if i tell you that both times we have got heads then these two heads this is sample space

and event is that we got heads both times. So students here we have only one outcome and this will be called a simple event. and now if we discuss the second type of event that is the compound event

if there is more than one possible outcome for it, right? In that case, this is called a compound event. For example, if I am saying that I have tossed a coin two times, So what is the probability of getting one head in a toss?

So the event that we will have students what will we have because we are getting the head in only one case exactly one time head right. when we get head exactly once, we will get this. So, this event that we will get will be the compound event. So, when we have more than one

outcome for any of the event then students that will be called a compound event or we can say that if any event is not simple. then that will be

compound event. Is that clear? Let's move on. Now see here, what does mutually exclusive event mean? Let me give you a small example. Okay, we have ticket numbers from one to twenty. We put them in a box, right?

There are ticket numbers from one to twenty in a box. There's a box like a lottery system. What did we do? from one to twenty? and tickets are kept inside this box.

whenever a person. will take a ticket. Either that ticket number is an even number.

or that ticket is odd. then what will its probability be? So when we look at outcome of even and odd, So students when we want it to be even then we will have two four

six, eight, right? So, these are the total possible outcomes that we can get. The second event is odd. So, in odd what will we get? One, three, five, seven, right? It's like this.

and this will go up to nineteen. So students, in both of these we are not getting any common event, right? So, you will see that A intersection B is coming to be empty. When we get this as empty,

then that is called as a mutually exclusive event. But students, if I say to you that we have a coin and that coin is being tossed. So tell me.

the outcome of that it is either a multiple of four. or it is a multiple of five.

then students will find the multiples of four. So the multiples of four are four. eight, twelve, sixteen, twenty. These are the multiples. What will be the multiples of five? Five, ten, fifteen, twenty. So students, you can see here.

A intersection B. In this we are getting what? We get twenty right? So, this means that this A intersection B that we get, it is non-empty.

this event that we get here students these two events will not be. Mutually exclusive. It states if two events A and B are said to be mutually exclusive, even if the occurrence of any one of them excludes the occurrence of the other event,

They cannot occur simultaneously. It means here the outcome is not the same in both the cases. So, this means that the intersection that is to be done here it should be empty Is that clear? If that is the case, then they both will be mutually exclusive events, right? Let's move on.

What is the meaning of exhaustive event? So, I just gave you the example of ticket numbers that from one to twenty we have ticket numbers in a box, right? So students, I have made a drawing of ticket here in front of you.

Tell me, is the number even or odd? If we take even numbers then it will be two, four, six till twenty. When we will take odd numbers, it will be one, three, five, seven and until nineteen. But when we take its union,

When we take the union of these two, it will create the complete sample space. This all will be the sample space. This means that both of these events are the exhaustive events. whenever we will get the events for these possible outcomes.

or whatever possible events that we get. When we'll take the union of all those events again, it creates the complete sample space. If that happens, then what is it called? It is called

exhaustive event. We throw a die and we get outcomes from one to six. This is even and this is odd but when we take its union we get s. This means that e one and e two it will be exhaustive event. We have given more examples of this. You all can easily

understand it. So, there is one exhaustive and mutually exclusive Students I just gave you this example. If we look here, this A is representing the even outcomes from one to twenty ticket numbers.

B is representing the odd outcomes. If we take its union, it gives complete sample space. But when we take its intersection, then we'll get this as empty. It is because there is no common event.

So, this is becoming both exhaustive and mutually exclusive, right? So students whenever we have any event. which is exhaustive and mutually exclusive.

then in that case what we have is called exhaustive and mutually exclusive event, right? What is the meaning of a complementary event? As I was discussing with you just now by using this example, here we have ticket numbers from one to twenty. So A.

is showing even number and B is showing odd number. and students A is the complement of B right. So, if A is not there, then this will be here. If not this, then this will come.

Complement is the complement of any set. For example, if I say the complement of A, what does that mean? Minus A from the sample space. minus A from the sample space. So, if we minus A from the sample space then what will we get?

will get B. That means the complement of A that we will get It is B. Students, what is this called? It is called the compliment event. So, you can understand this here right.

Take this example here, if we have two coins and we are tossing them It's head, head, head, tail, tail, head or tail, tail. if we say that.

We want at least one head, then what will be the possibilities of it? Head, head, head, tail and tail, head. and what will be the complement of this leaving aside these three what is left is tail tail So, this will be its complement from the sample space minus these three.

So what will we get? It will be called the complement event. And what does independent event mean? So, a set of events is said to be independent if the happening of any one of the events does not affect the happening of others. So, if by chance any one of the event is happening here.

that should not affect the other event. If something like that happens, then what are those events called? Independent events look an example given.

Suppose a coin is tossed twice, the event of occurrence of head in first throw. and event of occurrence of head in second throw are independent events. When coin is tossed two times, the fact that head appeared on the first toss has no relation to whether a head or tail will appear on the second toss.

We are throwing a dice and we roll it for two times. It is not necessary that if we get six for first time then we will get the same number second time, right? we can get other number. So first time we roll a dice and the second time we roll it, the outcomes in both situations will be independent.

There is no relationship between. two events. But you'll think, when will it not be independent? Let's take an example. There are total fifty two cards and from those fifty two cards, two cards are taken out.

and that card it is a red card. Right? Two cards were drawn and they were red cards.

Now again we'll draw two more cards. If it's red card again, then its probability will be affected because two cards were already drawn. When we draw the next two cards, then their probability will depend on the previous cards.

It is possible. It is because the cards and that two red cards from them have become fewer, right? Students, you need to pay attention to this point. The probability that we get here it will be dependent and the event also it will become dependent, right? So then whatever action is being done here, it is affecting the result. Let's take another small example for this. Let's say we have some balls.

here it is being said that We have four red balls. and

then we have four Blue balls. Now someone is taking out a ball, right? Let's say

that a person took out a ball. The ball that was drawn, it was blue. So blue ball is taken out. Next, when he goes to take out the second ball,

The possibility of it being blue is less and the possibility of it being red is more, right? It's because already a blue ball is taken out. But if this blue ball that is taken out, it is back inside.

and then pick another ball. One is with replacement, one is without replacement. So, when we did it with replacement, when ball was put back inside. and then the second ball is taken out, then events will be independent.

but if that ball is kept outside only and second ball is taken out then students in that case, the event will be dependent because it was done without the replacement, right? So, therefore,

Keep this in mind. Let's do more questions for better understanding. Two dice are rolled. A is the event that The sum of the numbers shown on the two dice is five.

And B is the event that at least one of the dice shows up a three. So, are the events A and B mutually exclusive or exhaustive? So, students when we are throwing any dice here.

then it is being told that we have a total of thirty six sample points. In that case, we have an event A that is the Some of the numbers shown on the two dice is five, right?

Students, the sum is five. And how can five be obtained? Either we have one and four as the outcome, right? or the outcome is two and three. or outcome we get is three and two or the outcome is four and one. So here, these are all the possibilities, right?

so. In these cases, what will the sum be? It's five. Now, what is information for B?

Students B is stating that B is the event that at least one of the dice shows up a three. It means one dice should show a three right so if one dice shows three At least one dice shows a three, so here we could have three one.

Next it could be three two. Next can be three three. This could be three four. This could be three five.

This could be three six. Now this was shown on the first one. Now we show it on the second one. So, it is one three right this could be two three. We have already taken three, so we will not take three again.

this will be four three this will be five three and this will be six three Now, see if there is any common element in these two. Students, what should be there in mutually exclusive? What should be A intersection B?

So, A intersection B. should be empty right. So, is there any common element?

Students, if we look, then A intersection B is it. Giving any value? Yes, absolutely it is giving a value. we are getting common outcomes. It is two three.

Yes, we are getting them both right. three two and two three are common. This means that it's non-empty, it is not empty. And this means it will not be mutually exclusive.

So, what will it be A and B? they are Not mutually exclusive, right? Students A and B, they both are not mutually exclusive first.

Second is exhaustive. Exhaustive means if A and B are union, then entire sample space is formed. There are thirty six points. Here we have four points and here we have two plus two and five that is we have a total of eleven points right.

when we take A union B students when A union B is taken So, what will we get from this? We will get one comma four, then we get four comma one. then the rest we will get three comma one right then three comma two right three comma three

Then students, we will get three comma four, right? then we will get three comma five. Then, three comma six. Then, one comma three, right?

Then students, we will get two comma three and four comma three. then we will get five three, then we get six three. And please note that this is not equal to the sample space.

This means A and B are what will the answer be for the second A and B. Both of them. They are not exhaustive, right? Is that clear?

So students, both of them will not be exhaustive. Got it? in this way we can check. Now let's look what are they saying next. A coin.

is tossed. So if the coin shows a head. Then we will draw a ball from a bag consisting of three blue and four white balls.

If it shows tail, we throw a die. Describe the sample space of the experiment. We have to show how this will be done. So here's what we have. Now pay attention. We will toss a coin first, right? students we have a box.

And we have one more thing here. What is it? Dice, right? Pay attention. and then we are given a dice. Is that clear? Now what is being said here is, if it shows head.

we draw a ball right if we have any sample space here then we have both head and tail right And students first, if we have event E here, the first event we have is that if this shows head. So what does it show? If it shows head, then we will draw a ball from a bag consisting of three blue balls and four white balls, right?

So students, here we will have three blue. and Four white balls, right?

so when we will get head here then in that case what will we do then we will take out the ball out from this So when we take out the ball, what are the possible outcomes? When we talk about this, what event can we get? So students, it can be. When we get head then the first ball will be blue.

or if we get head then students we will get the second one as blue ball. If we get head then the third ball is blue ball or if we get head then we get the first white ball. If we get head then second is white ball or the third one is white ball.

if we get head, then fourth one is a white ball. So these are possible outcomes. and let's suppose if we get tail such event can also happen that we get tail let's take this as e one this is e two Let us say we get tail. When we get tail, we will throw the dice here, right?

If it is tails, we throw a dice, right? So what could be the second sample space when we get this? So when we roll a dice, we can get tails as one. We can get tail as two or tails as three. We can get tail as four. We can get tail as five.

and we can get Tail as six, right? So here, what will we get? We will get the possible outcomes, right? Students, thus, whenever we create sample space, we will create it this way.

All right, now let's move on to the next question. from a group of three boys and two girls. Two persons are selected. Describe the sample space.

Students, what are we given here? We are given three boys and what else? and two girls. You are being told that we are selecting two persons. So then what are the possibilities we will get here and what event will be formed so that it could be both boys right.

Take B one and B two like that or maybe we get B one B three. Or maybe... we can get b two and b three right.

or we can get the first boy B one and second girl G-two or we can get second boy B-two and first girl G-one. or it can be third boy B-three and first girl G-one.

And students, we could have the first boy and the second girl, right? than the second boy and the second girl, right? and we can get the third boy B-three and second girl G-two.

we could get all this here it can also be first girl g one second girl g two right and besides that we won't have any other possible outcomes here right So in this way, we can make sample space here. It means we can make the sample space an event. And if we are asked whether both are girls or the first is a boy and the second is a girl, if anything is asked like that then we can create it here.

Students, here's another question. It says the experiment involves rolling a pair of dice and recording the number that come up. Describe the following event. so it is being said that the sum is greater than eight so what can A be if we are talking about greater than eight here It's greater than eight. It means when we add it, the total must be greater than eight.

So, students we can get this as a three comma six. or we can get four comma five right adding them will be giving us nine or students we can get five comma four right.

or maybe we can get six comma three now adding them the total is nine now ten So, if we do ten what will we get? It will be four comma six. then we will get five comma five and then we will get six comma four.

so this was for ten now we can also get eleven and students for eleven will get five comma six Right? we will get six comma five and six comma six. So these all will be

the outcomes. Where what will be the sum? It should be more than eight and this will become its event. Then the question states two occurs on the either die, right? The two should appear on one of the two dice. In that case, what could the sample space be? It could be two comma one.

two comma two. two comma three. two comma four.

two comma five. two comma six. or the other possible outcome is one comma two, right?

two comma two is already there so this will be three comma two. four comma two.

five comma two. six comma two What will we get here? It will be the value. The sum is at least seven.

and the multiple of three. What is being said? So students the sum it should be at least seven and it should be a multiple of three. What could it be? It could be nine or it could be twelve, right?

So these are the possible outcomes. See the event that we have. students what will we get? So what could event C be? It should be multiple of three and should be at least seven. It means

that we can get nine here. And when we use nine, what will we get? Then,

Here, we will get three comma six Then we will get four comma five. then we will get five comma four.

and this is six three. One possibility is this and another is twelve, right? So this is... at least more than seven and should be a multiple of three, right?

So, all multiples are there. So, these will be the possible outcomes. Question is whether A and B are mutually exclusive. We can see A and B don't have anything in common.

And in A and B, two is not present anywhere. In A, two is not present, right? and in B, two is appearing everywhere. So, in this, the A intersection B will be, if you see, the A intersection B that we will get will be phi. Hence, this is mutually exclusive. Now we need to see if B and C are mutually exclusive or not. So here we get two everywhere, right? But here we are not getting two.

So students B and C will be what? They both will be mutually exclusive. Next is whether A and C are mutually exclusive. So what can be common between A and C? So students, six, three is here also.

and then here also what else we get we also get five four here and here we get four five here and here also then six three is also here and here we get three comma six here So, what will we get here? The intersection of A and C that will be non-empty.

it means. We will have five points in common and students. intersection that we get is not empty. This means that A and C will not be mutually exclusive.

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