noc21- ma35 - lec10

[Music] [Music] hello friends welcome to the course essentials of data science with r software in which we are going to understand the topics of probability theory and statistical inference now you can recall that in the last lecture we had understood the basic elements of probability theory that how to define an experiment event etc once you have defined the events then we are also interested in different combinations of those events like if if this event and this event occurred together or only an intersection of those events occurred together and so on and this type of feature is very common when you are trying to use the data science so my objective in this lecture is simply to introduce you with these type of events where we are trying to take a combination of the events in different ways right now when i am trying to do it then obviously means at the end we are going to find out the probability of such an event but before that we have to understand that how we can define such events and when we are trying to define such events the vein diagrams that we have studied in mathematics they helps us a lot so it is possible that when you are trying to deal with the complicated problem that many times this vein diagrams and set theory day helps us so in this lecture i will aim that i define such an event and i try to represent those events in the format of vein diagram using the concepts of set theory i believe that theory you have done in class 10 or class 12 also so i am just going to use them and i will try to define different types of events and after that definitely our basic objective is that how to compute the probabilities of such an event and then how to compute them in the r software that will be our final aim so but at this moment let us try to prepare the foundations first so let us begin our lecture and and i believe that you have understood the meaning of an experiment or a simple event so now we try to define first an event in the format of union of events what does this mean suppose there are two events which i am trying to indicate here by capital a and capital b and obviously these two events are from the sample space capital omega now i define a new event which is a union b and the meaning of a union b is that a union b consists of all the outcome that are either in a or in b or in both a and b so now onwards we will call this symbol a union b as the union of the events a and b so this you can see that this is trying to define an event which is based on the combinations of the outcomes of events a and b for example in case if i take a very simple example suppose an experiment is conducted on the two simple experiments that is a baby is born in a hospital and we try to determine the gender of the baby that can be either male or female so now the sample space will consist of two events capital m and capital f where capital m is indicating the male child and capital f is indicating the female child and now let us try to define here the event a to b that the child is a male that is a boy and similarly we define another event b which is f that is b is the event that the child is a a female girl now since there are only two simple events here a and b so this a union b will be the collection of m n and f so in this case you will see that this a union b is the whole sample is space omega and we had learned in the earlier lecture that this event omega is equal to a union b in this case is called as sure event that definitely the child will be either male or female and now similarly we can also define the intersection of events so suppose a and b are the two events which belongs to the sample space capital omega so if we define here our new event a intersection b so this is called as the intersection of the events a and b this a intersection b consists of all such outcomes that are in both a and b so obviously the intersection a and b will occur only if both a and b occur right so this is how we can define the a union b and a intersection b and now we will try to view these events using thus vein diagram from the cut theory right so you know that when we are trying to create the vein diagrams we have this type of circles like this much this much or this and this like this right so this vein diagram will help us in visualizing the outcomes of different events a and b and in particularly when they are trying to occur together right so now using this vein diagram first let me try to define a union b and a intersection b and after that we will define different types of events so what we are going to do that we are going to uh to observe here two types of vein diagrams in which there will be some overlapping circles and in some cases there will be some separated circles so first we try to understand the vein diagram for a union b so suppose we have two events here a and b and these events are indicated by these two circles so circle number one is indicating the event a and circle number two is indicating the event b right so now a union b is the set of all such events a and b which occur when a simple event a or b occurs a or b occurs means if a occurs if b occurs like this and if both a and b occurs right so this entire gray shaded area that is going to indicate the area that is covered by the event a union b and you have to always keep in mind that when we are trying to use the word o r then from the statistical point of view a or b it means events a or even b this means that either a simple even from a occurs or a simple even from b occurs or a simple event which is a part of both a and b occurs right so this is how we try to interpret it and similarly when we try to look at the intersection b so once again i will use the circle number 1 to indicate the event a and circle number 2 to indicate the event b then a intersection b is the set of all simple events of a and b which occurs when the simple events of a and b occur together so if you can see here a is occurring here b is occurring here and this is the part where a and b are occurring together right so from the statistical point of view when we use the word and mean that means event a and even b occur that means both simple events from a and from b occur together and you will see that when we are trying to use this symbol a intersection b so in many books it is also written as a b a union b will be written always as a union b but a intersection b at many places you will find that this is written as a b so that is what you have to understand that is the simple terminology or symbolic symbolic quotations when we are trying to deal with probability theory now similarly we have defined some other events for example just like you have used the events a and b to define their joint occurrence together or separately so similarly we can define some other type of events for example we know from the set theory when we write a minus b that means all the events of a which are not in b so similarly i can translate this definition into the event and i can define this a minus b as an event that contains all simple events of a which are not contained in b you can see here this is the area grey area which is containing all the events of a which are not containing in the right so this shaded area is going to indicate the event a minus b right so this event a minus b or a but not b the quiz occurs if a occurs but b does not occur and if you try to represent this phenomena using the set theory then it can be represented by like this a minus b is equal to a union b complement right well that is clear from the vein diagram also and i am sure that you have done this thing in your earlier classes also and similarly i try to define here another type of event which is a complement a complement mean whatever is occurring in a except that everything occurs right so this a complement event we are trying to denote by a bar yes sometime you will see that in some books it is also defined at a c means c in the super script for this event a complement contains all simple events of omega which are not contained in a so in case if you try to represent this phenomena using the vein diagram you can see here this is your here capital omega right now you are trying to say this part here is a so in case if you try to ignore this part a then whatever is remaining this is omega minus a and omega minus a is your simply your here a complement so a compliment is actually omega minus a so this means the complementary event of a which is not a or a complement occurs whenever a does not occur and this entire gray area that is indicating all the events which are covered under the category of a complement right similarly we try to define here one more event a is a subset of b and which is indicated by this symbol that is a standard simple so the meaning of a is a subset of b is that all simple events of a are also part of the sample space of b right so you can see here that in this case this white circle this is the event b and this gray circle this is the event a right so this area that is defined here that is the part where the simple event of a this is also a part of b right so this is how you can see that different types of events are occurring and you can correlate these events with what is really happening in the real world and in order to give you an idea let me try to take here a very simple example to explain you that how these events can be defined so let me take a very popular example of rolling a die so we know that if our die is rolled once then the possible outcomes are going to be the number of dots on the upper surface of the dice like this one if this is your hair dyes then whatever is occurring here suppose 3 is occurring that means this is our outcome right and there are six such possibilities of getting numbers one two three four five and six so now when we try to define here the sample space so sample space will consist of all the simple events and we have indicated that the simple events are going to be indicated by the symbol small omega so we try to take here omega 1 is equal to 1 that means number 1 occurs on the upper surface omega 2 is the number 2 occurs omega 3 is the number 3 occurs omega 4 is the number 4 occurs omega 5 is the number 5 occurs and omega 6 is the number 6 occurs so this omega capital omega the sample space will consist of 6 points omega 1 omega 2 omega 3 omega 4 omega 5 and omega 6. now in case if you try to define here an event which consists of 5 numbers 1 2 3 4 5 then this event a can be defined as the collection of points omega 1 omega 2 omega 3 omega 4 and omega 5 and we define one more event here b which is the set of all odd numbers so you can see from this capital omega the odd numbers are occurring with 1 3 and 5. so that means b is going to contain the simple events omega 1 omega 3 and omega 5 and you can see here that this elements omega 1 omega 3 and omega 5 they are here right so obviously b is a subset of a so this is how you define such an event right and in case if you try to think about a real life scenario suppose a shopping website is being accessed by both male and females and they have the entire observations and they want to study that which of the clothings are going to be sold more for male or for females so you have to collect all the data and then you have to take out the subset of the females or male from the entire data set so in those types of events this type of events and their probability are going to help you now i try to take one more example in which i try to define the event a as consisting of three numbers 2 4 and 6 which are actually even numbers out of this capital omega so we have a three simple events omega 2 omega 4 and omega 6 in the event a and then we try to define here one more event b which consists of all the numbers which are divisible by 3. so there are two number 3 and 6 which are divisible by 3 so we try to create another event here b which is the collection of omega 3 and omega 6. now in case if you try to define here a union b that means all those events which are in both a and b that mean this will consist of omega 2 omega 4 omega 6 this is here omega 2 omega 4 omega 6 and then you have here omega 3 and omega 6 from event b so this omega 3 is here and omega 6 is here which is already here so this a union b will consist of all the points or the simple even which are either even or divisible by 3 or both so this will consist of 4 numbers omega 2 is equal to 2 omega 3 is equal to 3 omega 4 is equal to 4 and omega 6 equal to 6. so this is how you can define here the union of the events now i take one more example and right try to create an event here capital a which is the set of all the odd numbers 1 3 and 5 out of capital omega so this will be having three sample points omega 1 omega 3 and omega 5 and we try to define here the same event b which consists of those numbers which are divisible by 3. so there are two number 3 and 6 which are divisible by 3 so this b is the collection of two points omega 3 and omega c now i try to define here the event a intersection b so a intersection b is going to contain the common points between a and b so you can see here that omega 3 and omega 3 are the two point which are common in both the a and b so this a intersection b event is the set of the simple event in which the numbers are odd and they are divisible by 3 right well in case if you try to add here one more number here suppose omega 6 then what will happen then this a intersection b will consist of two points omega 3 and omega 6 but still the omega 6 will not happen here because you have a condition here that the numbers have to be odd that is what you have to keep in mind that when you are trying to define different types of combination you have to check and verify whether these things are happening or not right okay so now in case if you try to take one more example to define a minus b so we consider here an event a which consists of omega 1 omega 3 and omega 5 which are the set of all the odd numbers as earlier and we try to define here this is the same event here b consisting of 2.3 and 6 which is the set of number which are divisible by 3. then a minus b is going to be defined by the collection of those point which are in a but not in b if you try to see here omega 1 which is not in b so it comes over here now you try to see that omega 3 is common to both so omega 3 will not appear in a minus b and similarly if you try to see here this omega 5 it is also not here in this capital b so omega 5 will appear in the event a minus b and we are not bothered about omega 6 right so this a minus b event will consist of two points omega 1 and omega 5 which is essentially the set of simple events in which the numbers are odd but not divisible by 3. so you can see you have obtained what exactly you wanted depending on your objective right now similarly i try to define a complementary event so if i try to define here an event a which is the set of even numbers so there are three numbers 2 4 and 6 in capital omega which are divisible by 2 the other even numbers so whatever numbers which are in a they will not occur in a bar so omega is 1 2 3 4 5 and here 6. and now we try to remove this points a that is 2 4 and here 6. so now what is here left this is a the numbers 1 3 and 5. so this a complement is going to have all the three numbers omega 1 omega 3 and omega 5 which are essentially the set of odd numbers and obviously if you try to see a union a complement this is going to be omega and that you can see here from these two events also and even if you try to see from this vein diagram also this is your here capital omega consists of the point 1 2 3 4 5 6 which is here like this for this omega and now you are trying to have here a circle indicating the event a which is consisting of omega 2 omega 4 and omega 6. so now this a complement is consisting of this part here which is the dotted part here which is here omega 1 omega 3 and omega 5. now if you try to combine this omega 1 omega 3 omega 5 and this omega 2 omega 4 omega 6 that is going to give you the complete omega that is the point 1 2 3 4 5 and 6. right so whatever you have done here you can verify them through the wind diagrams also now i try to define one more property of this events two events a and b are said to be disjoint if a intersection b is equal to five that means there is no element which is common between a and b and if you try to create the vein diagram of such an event this will look like this right this is your here capital omega and so you can see here there are no points which are common between this a and b and if they are common the possibly the vein diagram will look like this right or that may be something like this also so the interpretation of disjoint event is that both the events a and b they cannot occur simultaneously so you can now correlate this statement with the happening of various types of events you want to compute the probability right and in case if you try to see here from yours even point of view then the event a and a complement both are disjoint event right for example if you want to see in through the vein diagram this is your here a and this part here is your a complement so if i try to make this a to b here by dotted then this a complement is the remaining parts and you can see here both of them cannot occur together right okay now whatever i have told you about these events i have explained you by taking the example of only two events just for the sake of simplicity so that you can understand it easily but these events can be be defined for more than two events also these definitions whatever you have done either for union intersection etc they can be extended for more than two events also so let me try to give you a fair idea about these things suppose if i say i have here more than two events which are indicated by a one a two a m so now we have here m events now these events are said to be mutually or pairwise is joint if a i intersection aj is equal to phi and in this case yeah whenever i is not equal to j and and it goes from 1 to m and let me try to show you this definition by this example also for example the same example in which the die is roll and we have here six points in the sample space if i try to take here two events here a and b like as a is the collection of all those points which are odd on and b is the collection of all those points which are even in numbers then these both these events a and b both are disjoint because either it can occur and when a is occurring then no number which is even can come and similarly if b is occurring that means only the even number are coming to myspace so this odd numbers cannot come so this is how both this events a and b they are going to be disjoint right and what i am trying to say here that if you want to know about that more than two events are going to be disjoint or not suppose if i say that there are three events a b and c and if i want to know whether they are disjoint or not so we have to check their property pairwise i have to check for a and b b and c and a and c right and if all of them are equal to phi then i can say that all the events a and b and they are mutually disjoint so now similarly we can extend the definition of unions means earlier we have understood what is the meaning of a union b so similarly instead of having only two even suppose i have here more than two even say an events and they are indicated by a one a two a m so the union of a1 a2 am it is going to be indicated by a1 union a2 union up to am and this is also an event and this event is defined to be the event consisting of all the outcomes that are in a i for at least one i i goes from 1 to m or in very simple words i can say that the union of the ai occurs when at least one of the events ai occurs that is as simple as that and this is simply an extension of the concept that you have learnt in the case of a union b right now as we have defined the union of more than two even similarly i can also define the intersection of more than two events so suppose if i have here m events say a1 a2 am then the intersection of these event is going to be defined as here a1 intersection a2 intersection am and this event is defined to be the even consisting of those outcome that are in all events a1 a2 am right and in very simple words i can say that the intersection of these events occurs when all the events ai occurs right so this is how you can define here the the intersection of more than two events also ok so now we come to an end to this lecture and you can see that this lecture was a very simple lecture and but it was trying to correlate what you have done earlier in your set theory and in vein diagrams and through this thing i am trying to interconnect the probability theory whenever you are trying to compute any probability this is obvious that you need to know that what is the event what is your objective what you really want to find and based on that you have to define an event in a proper way and then you have to see how this event is going to occur there can be different ways of knowing it either you can compute the number of events the number of possible events which are going to happen or not happen and exactly on the same line this vein diagram will also help you in counting of those numbers which are favorable to us and it will be your choice that which of the methodology you are going to opt when you are trying to implement these concepts in real life so now i would request you why don't you take some examples from your book from your set theory and vein diagram and try to just practice and have an idea what is the meaning of the union intersection complement minus etcetera once you practice it for simple events like as two events or three events it will not be difficult for you to extend this definition for a general case so you try to practice it and i will see you in the next lecture with more details on probability theory till then good bye [Music] [Music] you