 Assalamu alaikum. Welcome to lecture number 16 of the course on statistics and probability. Students, today is the 16th lecture and the beginning of the second part of this course. As I mentioned in the very first lecture, the first 15 or so lectures were concerned with descriptive statistics. The next 15 are going to be concerned with probability theory and the last 15 will pertain to inferential statistics. So, during the first 15 lectures, you have learnt different techniques of summarizing and describing data, data that you collect on sample basis. Our general discussion was about the univariate situation in which we were describing a single variable in different ways. And in case of a quantitative variable, we discussed at length the frequency distribution of the variable and its shape, its spread and its measure of central tendency. Also in the last lecture, I discussed with you a very important concept and that was regression and correlation when we tried to relate two variables with each other. I hope that you enjoyed learning the various concepts that were discussed in the first part of the course and I hope that you will enjoy probability theory as much if not more. You will say probability as a difficult concept we can enjoy. Actually students, this is not the case. This is an impression that is all around that it is supposed to be the most difficult topic. The only thing is that you need to have a methodological approach. If you approach it in a systematic way, then you will see that it is not really that difficult and in fact it is quite enjoyable. In this part of the course, we will begin with the basic concepts of probability and we will go on to discuss discrete and continuous probability distributions. In particular, we will be discussing the binomial distribution, the hypergeometric and the Poisson distributions and also the most important distribution in statistical theory, the normal distribution. So, let us begin with the very basics. First of all, let us discuss the definition of probability. What do you understand by the word probability? I am sure that you will reply that when we are talking about chance, that is what we mean by probability. We are talking about the chance of something, the chance of something. But in the next few lectures, you will see that there are various ways in which you can define probability. We have the classical definition, we have the relative frequency definition and then we also have the subjective or the personalistic definition of probability. These are definitions, we will discuss one after another and you will realize that the most important one from the statistical point of view is not the subjective, not even the classical, but the relative frequency definition of probability. Because this is the definition that pertains to real life phenomena and also a definition which enables you to quantify probability. You can express it in a number form, for example, a particular event probability is 75 percent and another event probability is 55 percent. We will discuss these things in detail during these 15 lectures, but today we have to start from the very basics. Students, the first thing that we will discuss in this regard is set theory. It will simply be a review of what you have already studied and the reason why we are going to discuss it is that as you will see later, there are a number of concepts of probability theory which are facilitated by the use of set theory. Let us start from the definition of a set. As you all know, a set is a well-defined collection or list of distinct objects. For example, a group of students, the books in a library, the integers between 1 and 100, all human beings on earth, etc. etc. In its definition, I use two words, well-defined collection of objects and distinct objects. What do these two mean? Well-defined means that I should be absolutely clear as to whether a particular object belongs to or not belongs to a particular set or distinct se yehi murad hai that any particular element or object should appear in that set once and only once. Students, the set itself is denoted by a capital letter usually such as capital A, capital B, capital C and the elements in the set are denoted by small letters. As you see in the examples on the screen, the set A could be consisting of small a, small b, small c and small d and the set capital B may consist of the numbers 1, 2, 3 and 7. The number of elements in a set A, this is denoted by n A and it is called the number of the set A. As you see on the screen, if a set consists of 4 elements then we will say that the number of this set is 4 and we will write n A is equal to 4. If x is an element of a set A, we write x belongs to A and if x does not belong to A then we write it with the same notation but crossing on the notation to denote that x does not belong to A. As you can appreciate a set can have any number of elements and if a set does not have even a one single element it is called an empty set or a null set and it is denoted by the Greek letter phi. But students please note that the set consisting of the element 0 is not a null set because it contains one element and that is 0. In fact, if a set contains only one element we say that it is a unit set or a singleton set. But students, if we are talking about an element x then we will not put a parenthesis or bracket but if we are talking about a set containing one element x then we will be putting the parenthesis around the letter x as you now see on the screen. A set may be specified in two ways the first is called the roster method and in this method we give a list of all the elements of a set. For example, if we throw a die our set of all possible outcomes consists of 1, 2, 3, 4, 5 and 6 and if we toss a coin our set consists of head and tail. The other method is called the set builder method or the rule method and in this method we state a rule that enables us to determine whether or not a given object is a member of our set. For example, if we write that a is the set of all x values such that x is an odd number and x is less than 12 this means that we are talking about the set 1, 3, 5, 7, 9 and 11. Speaking of sets it should be kept in mind that the repetition of an element or the change of order of the elements in a set does not alter the set meaning that if I write 1, 3, 5 that is the same set as 5, 3, 1 or 5, 1, 3. The size of a set is given by the number of elements present in it. The number NA in other words denotes the size of a set. This number may be finite or infinite. A set is finite when it contains a finite number of elements otherwise it is called an infinite set. The empty set is regarded as a finite set. Examples of finite sets would be the set A of consisting of all the positive integers from 1 to 100. The set B consisting of x values where x represents a month of the year, C representing those x values which represent printing mistakes in a book and D representing those x values which represent living citizens of Pakistan. In the examples may agree that we are talking about finite sets because number of living citizens of Pakistan is not an infinite number. Number of mistakes in a book cannot be an infinite number and so on. On the other hand examples of infinite sets are the set of even integers or the set of all real numbers between 0 and 1 including 0 and 1 or the set of points on a line or the set of the sentences in the English language. Of course it can be argued that the number of sentences in the English language is not an infinite number but the point to understand is that sometimes if the set or if a population is very very large so large that for practical purposes we can regard it as infinite then we do adopt this strategy. On the other hand in the first three examples it is obvious that they are really infinite sets because the total number of even integers or the number of real numbers all the real numbers between 0 and 1 or the number of points on a line they cannot be regarded as finite entities. These four examples that I have just given you will note that the first three examples were the set of all points on a line or the set of even integers or the set of all numbers between 0 and 1 inclusive because they you can simply not count them as finite entities. But the last example the set of all sentences of the English language it cannot be an infinite number truly speaking. So, sometimes when sets or samples or populations if they are so large that for all practical purposes they can be regarded as infinite then we do adopt this concept. A set that consists of some elements of another set is called a subset of that set. If B is a subset of A then every member of set B is also a member of set A. For example if set A consists of the elements 1 2 3 4 5 and 10 and B consists of 1 3 and 5 then B is a subset of A in other words B is contained in A. Speaking of subsets it should be noted that every set is regarded as a subset of itself and the null set phi is regarded as a subset of every set. Two sets A and B are equal or identical if and only if they contain exactly the same elements. In other words set A is equal to set B if and only if A is a subset of B and B is a subset of A. Speaking of subsets we also need to differentiate between the proper subset and the improper subset. As you see on the screen if a set B contains some but not all of the elements of another set A while A contains each element of B. In other words if B is contained in A and B is unequal to A then the set B is defined to be a proper subset of A. A very important concept is that of the universal set that set of which all other sets are subsets is called the universal set. As you now see on the screen it is the set which contains all possible elements under consideration. It is also called the space and it is denoted by either by capital S or by capital omega. An interesting question is how many subsets can a set have? As you see on the screen a set S containing n elements will produce a totality of 2 raised to n subsets including the set S and the null set phi. For example, if we consider the set A consisting of the numbers 1, 2 and 3 this set will contain it will generate 2 raised to 3 that is 8 subsets and these subsets are phi 1, 2, 3, 1, 2, 1, 3, 2, 3 and 1, 2, 3. In the context of set theory a very important and useful concept is that of the Venn diagram. A diagram that represents sets by circular regions parts of circular regions or their compliments with respect to a rectangle representing the space S is called a Venn diagram. The Venn diagrams are used to represent sets and subsets in a pictorial way and to verify the relationship among the sets and the subsets. A simple Venn diagram is of the form that you now see on the screen and the one that you see at this time pertains to disjoint sets those two which do not have any element in common. On the contrary if there are two sets A and B such that they have a few elements in common then the Venn diagram is of the form that you now see on the screen. The next concept that we must concentrate on is the concept of operations that can be performed on sets. Sets can be combined in order to form new sets and the four main operations that we should consider are union, intersection, complimentation and set difference. The union or sum of two sets A and B means the set of all elements that belong to at least one of the sets A and B that is A union B is the set of all those x values which belong either to A or to B. For example, if set A consists of the numbers 1, 2, 3, 4 while set B consists of 3, 4, 5, 6 then A union B consists of the numbers 1, 2, 3, 4, 5 and 6 or the new set 1, 2, 3, 4, 5 and 6 students note that it is fulfilling the basic property that I just mentioned in terms of union. So, some of them belong to A but not to B, some of them belong to B but not to A and two of them belong to both A and B. So, we can say that the elements of this set belong to at least one of the two basic sets A and B. The intersection of two sets A and B means that set in which the elements belong to both A and B. For example, if set A consists of 1, 2, 3, 4 and set B consists of 3, 4, 5 and 6 then A intersection B is the set of the elements 3 and 4 as these are the two elements which belong to both A and B. Two sets A and B are defined to be disjoint or mutually exclusive or non-overlapping when they have no elements in common that is the intersection of the two sets is the null set 5. For example, if set A consists of all possible outcomes of a die that is 1, 2, 3, 4, 5 and 6 and set B consists of all possible outcomes when we throw a coin that is head and tail then students it is obvious that these are disjoint sets in the sense that they have no element in common. The difference of two sets A and B denoted by A minus B is the set of all elements of A which do not belong to B. Representing this point by the Venn diagram that you now see on the screen, the shaded area of the set A represents the set A minus B because it is all those elements of A which do not belong to B. And the last concept in this regard is that of complementation. The particular difference S minus A that is the set of all those elements of S which do not belong to A is called the complement of A and it is denoted either by A bar or by A C. Symbolically A bar is the set of all those x values which belong to S but which do not belong to A. In the Venn diagram that you now see the shaded portion of the universal set S represents the complement of A because it contains all those elements which are not contained in A. The next important point to be considered is the algebra of sets and this is that part of set theory which provides us with a number of laws that enable us to solve a number of problems. The first law is the commutative law and as you now see on the screen it is given by A union B is equal to B union A and A intersection B is equal to B intersection A. I think that you will agree that this law is quite self-evident. Zahir hai ke aapne A aur B ka agar union lena hai to aapne unko hi kathya karna hai to agar aap A union B kaheen ya B union A kaheen it is one and the same thing and the same holds for intersection. The next law is the associative law and it states that A union C is the same thing as A union B union C and similarly for intersection. The distributive law is given by A intersection B union C is equal to A intersection B union A intersection C. Also A union B intersection C is equal to A union B intersection A union C. Students iss law ko yaad rakhne ka aasan tari kaya hai ke aap ordinary multiplication or addition ka example apne zaheen merakheen. When we write A into B plus C we know that it is equal to A into B plus A into C. Isi tara se agar aap isko dekhenge to you will see that it is exactly the same pattern. The idempotent law is extremely obvious A union A is equal to A and A intersection A is also equal to A. The identity laws are A union S is equal to S A intersection S is equal to A. A union phi is A and A intersection phi is equal to phi. I would like to encourage you to draw the Venn diagram and to work with it yourself and to decide for yourself whether or not these are correct. Similarly, we have some other laws as you now see on the screen. The complementation laws are given by A union A bar is equal to S A intersection A bar is equal to phi. A double bar is equal to A S bar is equal to phi and phi bar is equal to S. Also we have the D Morgan's laws and these are given by A union B whole bar is equal to A bar intersection B bar and A intersection B whole bar is equal to A bar union B bar. Once again it is very simple to verify all these laws. Aap kuch numerical examples le lije take examples of sets which contain various numbers and then verify each and every one of these laws. The next concept is that of the partition of a set. And as you now see on the screen, a partition of a set S is a subdivision of the set into non-empty subsets that are disjoint and exhaustive that is their union is the set S itself. This implies that a i intersection a j is equal to phi where i is unequal to j and a 1 union a 2 union a 3 so on up to a n is equal to S. Let me explain this to you with the help of an example. Consider the set of all possible outcomes when we throw a die and that is 1 2 3 4 5 and 6. Now, if I partition this set into three parts in such a way that the first part consists of the elements 1 2, the second part consists of 3 4 and the last part consists of 5 6. Students this is a partition of the set S into three parts and it fulfills the formal definition that I just presented to you. The union of the three sets 1 2 3 4 and 5 6 is equal to the big set 1 2 3 4 5 and every one of these three sets is mutually exclusive 1 2 is different from 3 4 and 3 4 is different from 5 6 and there is no part of any of these three sets which is overlapping with any other set. The next concept is that of the class of sets. By a class of sets I mean a set of sets. For example, if you consider the set of lines this is a class of sets because each line itself is a set of points. The class of all subsets of a set A is called the power set of A and it is denoted by P of A. For example, if A is the set head and tail then the power set is given by the null set phi, the set H, the set T and the set HT. You will remember that a few minutes ago I told you that if a set consists of N elements then the total number of subsets that it can produce is 2 raised to N and we also considered an example there. So, the definition given under that example, the set that we considered of the 2 raised to N subsets of the set that set was also a power set. Another very important and interesting concept is that of the Cartesian product of sets. As you now see on the screen the Cartesian product of sets A and B denoted by A cross B is a set that contains all ordered pairs x y where x belongs to A and y belongs to B. For example, if the set A consists of HT and the set B consists of 1, 2, 3, 4, 5 and 6 then the Cartesian product A cross B is the set of the ordered pairs H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5 and T6. This silsle may important point here is that in general A cross B is not the same thing as B cross A because if you consider the graph in which x is on the x axis and y is on the y axis you will readily agree that 1 comma 2 is not the same point as 2 comma 1. Students I have discussed with you the basic concepts of set theory which I am sure that many of you are already familiar with, but it is good to revise and to refresh them in your mind because as you will see in the forthcoming lectures this theory will be very, very useful for us when we talk about probability and a number of probabilistic problems. Another important mathematical theory which we will be using in solving probabilistic problems is the theory of counting rules. There are three rules that enable us to solve a number of problems in a convenient manner and these are the rule of multiplication, the rule of permutations and the rule of combinations and I will pick them up one by one. As you now see on the screen the rule of multiplication is stated as follows. If a compound experiment consists of two experiments such that the first experiment has exactly M distinct outcomes and corresponding to each outcome of the first experiment there can be n distinct outcomes of the second one then the compound experiment has exactly M n outcomes. Students, this rule is extremely simple. Let me explain it to you with this rule. With the help of an example, let us go back to exactly the same one that we discussed a short while ago. One set consisting of h t and the other one consisting of 1, 2, 3, 4, 5 and 6. I am sure that this is the case of tossing a coin as well as tossing a die. The coin can result in two possible outcomes head and tail and so M is equal to 2 and the die can result in six possible outcomes and so n is equal to 6. Now according to the multiplication rule the total number of ways in which this compound experiment can be performed is M into n. So, in this case it is 2 into 6 and that is 12 and students as you will remember 12 is exactly the number of ordered pairs that we had when we considered this example earlier as you now see on the screen. The 12 ordered pairs are H 1, H 2, H 3, H 4, H 5, H 6, T 1, T 2, T 3, T 4, T 5 and T 6. Here is the example I have just discussed. This may be considered two simple experiments being performed together to give us the compound experiment, but students this can be extended to any number of simple experiments which are being conducted together to give us the compound experiment. And the other point is that this rule of multiplication theorem is also called the rule of multiple choice and the example that I will now consider and discuss with you will portray this point clearly. As you now see on the screen suppose that a restaurant offers three types of soups, four types of sandwiches and two types of desserts. Then a customer can order any one out of 3 into 4 into 2 that is 24 different meals. Let us consider the points that I just conveyed to you. This has three simple experiments. The choosing of one soup out of the three possible choices, the choosing of one sandwich out of the four possible sandwiches and the choosing of one dessert out of two. Why this rule is also called the rule of multiple choice. The choosing of the soup, the choosing of the sandwich and the choosing of the dessert. So we are extending the initial rule M n to the case of three experiments and we are now applying the rule M into N into P where M is the number of ways of choosing a soup, N is the number of ways of choosing a sandwich and P is the number of ways of choosing a dessert. Let us consider another very interesting example. Suppose that we have a combination lock on which there are eight rings. In how many ways can the lock be adjusted? This lock has eight positions and we are trying to determine how many different ways in which we can adjust or set this lock in order to lock the suitcase. Now students, how do we approach this problem? The logical way to go about it is to think of the eight positions and to realize that for any one of those eight positions it can be filled in ten different ways. There are ten digits from 0 to 9 or juphehli position hai uspe bhi 0 to 9 koi bhi digit hum rakh sakte hai. Same for the second position, same for the third and same for the eighth. So if we apply the rule of multiplication, we find as you now see on the screen that the total number of ways of doing this is ten into ten into ten and so on, so that the answer is one hundred million. Can you imagine one hundred million ways of adjusting that lock on your suitcase? One hundred million yani das karoor and the formula in this case when all eight positions have the same number of possible ways of being set, the formula can be simply stated as ten raised to eight, ten into ten into ten eight times. But fundamentally it is the same rule that I have been discussing for the past few minutes, the rule of multiplication. Students this brings us to the end of today's lecture and in the next lecture I will discuss with you the other two very important rules which enable us to solve a number of problems in probability theory, the rule of permutation and the rule of combination. After that we will proceed to the real thing itself and that is probability. Until next time, my best wishes to you and Allah Hafiz.