 As Salaamu Alaikum. Welcome to lecture number 19 of the course on statistics and probability. Students you will recall that in the last lecture I discussed with you various approaches to probability. We discussed the subjective approach and the objective approach. Under the objective approach I discussed with you in detail the classical definition of probability. Also I touched upon the relative frequency definition which is the very important definition from the statistical point of view. Today I will discuss with you this definition in a little more detail and then we will proceed to the more mathematical definition and that is the axiomatic definition of probability. As indicated in the last lecture the relative frequency definition pertains to those situations where we can say that a particular experiment has been repeated a large number of times. As you can see on the screen if a random experiment is repeated a large number of times say n times under identical conditions and if an event a is observed to occur m times then the probability of the event a is defined as the limit of the relative frequency m over n as n tends to infinity. Symbolically we write p of a is equal to the limit of m over n as n tends to infinity. Students in this definition we have nothing to do with the m and n of the classical definition. In classical definition n denoted the total number of possible outcomes of a random experiment and m was the number of the favorable outcomes. But here n denotes the number of times your experiment is being repeated and m denotes the number of times the outcome of your interest occurs. Actually if we want in this definition m or n ki bhajai hum koi aur letters istimal kar sakte the. Now the next thing is and the important point is that this definition assumes that the ratio m over n tends to us tends to become stable as n increases indefinitely. Let me explain this point to you with the help of an example. As you now see on the screen if we consider the coin tossing experiment no one can tell which way a coin will fall. But if it is a symmetrically constructed coin we expect the proportion of heads and tails after a large number of tosses to be nearly equal. An experiment to demonstrate this point was performed by Kerish in Denmark as far back as 1946. He tossed a coin 10,000 times and obtained all together 5,067 heads and 4,933 tails. The behavior of the proportion of heads throughout the experiment is shown in the figure that you now see on the screen. The x axis represents the number of tosses that were carried out whereas the y axis represents the proportion of heads that we obtained for various numbers of tosses. Students, you have noted that this diagram shows that in the beginning there were wide fluctuations. Sometimes the proportion of heads was much higher than 10.5 and sometimes it was much lower but as the number of tosses increased those fluctuations began to diminish and that curve representing the proportion of heads tend to the number 0.5. If you have a look at the diagram once again carefully you see that this trend of the proportion of heads approaching the value 0.5 is very clear as the number of tosses increases and it is quite easily understood that based on this trend that we see we can expect that if the number of tosses were to be increased indefinitely this proportion would tend to the limiting value 0.5. Students, this is the value that this proportion is tending to as n tends to infinity. This limiting value is taken as the probability of heads in this particular example. Let me share with you another very interesting example and this pertains to real life data that was collected in England. This example that I will be sharing with you is related to the sex ratio, the proportion of male births as compared with the female births and the first thing to note in this slide is that ever since the 18th century at least in certain parts of the world it has been noticed that in reliable birth statistics there is always a slight excess of boys as compared with girls. As you now see on the screen, Laplace recorded that among the 2,15,599 births in 30 districts of France during the years 1800 to 1802 there were 1,010,312 boys and 1,05,287 girls. The proportions of boys and girls were thus 0.512 and 0.488 respectively indicating a slight excess of boys over girls. Students, you have noted that the total number of births that were considered were 2,15,599 and that is a large number and the definition that I am discussing with you is that the total number of times the experiment is repeated that number should be large. 2,15,599 this is a fairly large number and if this proportion of male births is slightly higher compared with the female births then we can say that the probability of a male birth is slightly more than 0.5 and you may be a little surprised to hear this because I think generally we think that probability of a male birth and probability of a female birth they are both 50 percent. Of course, if the number of births were smaller if n is a smaller number then we can expect that the proportion of heads will be different from the limiting value as you noticed in the example that I discussed with you earlier in the coin tossing experiment when we had only a small number of tosses the proportion of heads was sometimes much higher and sometimes much lower than 0.5. So, let us consider another very interesting example as you see on the screen we now have a table in front of us showing the proportions of male births that had been worked out for the major regions of England as well as the rural districts of Dorset for the year 1956. If you concentrate on the column of the rural districts of Dorset you notice that the proportion of male births fluctuates between 0.38 and 0.59. In the district of Beaminster it is only 0.38 but in the district of Shaftesbury it is as high as 0.59. On the other hand if we have a look at the various regions of England which are obviously much larger in population than the rural districts of Dorset we find that the proportion of male births fluctuates between 0.512 and 0.517 the fluctuations for the various rural districts of Dorset are much more than the fluctuations for the major regions of England. And the point to be noted is that the figures for the various rural districts of Dorset are based on only 200 births or so each whereas the figures for the major regions of England are based on about 100,000 births each. So students the same point is validated when you have a small number of trials, a small number of repetitions of the experiment, the proportion of the event of your interest will fluctuate quite a lot and it will not be close generally to the limiting value. The value that it will approach only when the number of trials, the number of repetitions of the experiment increases to a large number such as 100,000 in this example. If you have a look at the table once again you notice that the proportion of male births for the entire country is 0.514 and the various proportions for the various regions of England do not deviate much from this overall proportion for the whole country. So I think you will agree that we can assume that the proportion tends to the limiting value 0.514 or something quite close to that because that is the proportion that is valid for the entire country. Though this example says that the probability of male birth according to this definition is higher than 0.5 and it is something close to 0.514. Students why is it that I said a short while ago that this particular definition is the most important definition from the statistical point of view. The reason is that in the real life situation, in the real life world there are numerous situations where we do not have a situation where the various possible outcomes of a random experiment are equally likely. Look how many times we toss a fair coin or a die in a real life. We cannot say that the thing is so symmetric or that situation is so perfect. That one outcome is as likely as the other. Rather we are faced with this kind of a situation where one outcome is more likely than the other and then there is no way of determining the probability of that event the one that we are interested in except by the definition that I just presented. If we are interested in determining the probabilities of this kind, what is the probability that the next student who will be enrolled in the virtual university BCS program is a first divisioner in his FA or FSC. What is the probability that the next man who walks into this room is a smoker? What is the probability that the next person who walks into this hospital has high blood pressure? Students, you have to be careful that to answer these questions you are not in a position to apply the classical definition of probability. Therefore, being a first divisioner is not as likely as not being a first divisioner. Similarly, being a smoker and not being a smoker, these are not equally likely. Being a person with high blood pressure and a person without high blood pressure and with normal blood pressure, they are not equally likely. To answer these questions you will have to resort to the relative frequency definition of probability. You will have to collect data from a large number of persons or objects if it is some objects that we are talking about. You will have to collect data from a large number of persons of that kind. If you are talking about students, then look at the record of the students enrolled in the virtual university BCS program and look at the proportion of first divisioners and then you might agree with me that you can treat that proportion as the probability that the student who is enrolled in the virtual university will be a first divisioner. In the last lecture and in today's lecture, I have discussed with you various approaches to probability. The diagram that you now see on the screen depicts the various approaches and the various definitions. We can say that probability is of two kinds, non-quantifiable and quantifiable. By non-quantifiable I mean the inductive approach, the subjective or the personalistic probability. These are different ways of talking about the same thing and you will recall the example that I presented to you in the last lecture that a panel of three judges is hearing a trial. Two of them decide on the basis of the evidence that is presented that the person who is being accused is guilty but one of them decides that the evidence is not strong enough for him to decide that this person is guilty. The quantifiable probability can be divided into two categories. The classical definition of probability is valid in the situation where the various possible outcomes are equally likely and the relative frequency definition applies in other situations where we cannot say that the various possible outcomes are equally likely. A statistician's main concern is this third definition of probability because in statistics by and large we are dealing with real life phenomena where various possible outcomes are not equally likely. The relative frequency definition says that the subjective probability is the one that we are encountering most of the time in our daily lives. As I said last time, we are taking such probabilistic decisions every now and then but the point of view of statistics cannot come to our work, that definition because students I think we can say that it is next to impossible to quantify the subjective probability. You see you can say that something is highly probable, something is moderately probable or something is very improbable in that context when we are doing it in a subjective manner. A particular painting is very beautiful or it is quite beautiful or it is not beautiful but you will not be able to say that the beauty value of the painting of Mona Lisa is 0.96 or 0.91. Though we are doing subjective probabilistic decisions in our mind all the time in our daily lives but for statistical analyses and for statistical research and all these scientific methods that we want to do to draw conclusions about various phenomena on the basis of real data, we would generally apply the relative frequency definition of probability. I have given you so many different definitions and so many details of probability but students the most mathematical definition is yet to come and this is the axiomatic definition of probability. This was formulated by the Russian mathematician Kolmogorov as far back as 1933 and as you now see on the screen this definition says that let S be a sample space of a random experiment with the sample points E1, E2 so on so on up to En. To each sample point we assign a real number denoted by the symbol P of EI and we call it the probability of EI if it satisfies the following three basic axioms. Axiom 1 for any outcome EI, P of EI is a number lying between 0 and 1. Axiom 2, P of S is equal to 1 where S represents the shore event and axiom 3 if A and B are mutually exclusive events in other words disjoint subsets of S then the probability of A union B is equal to the probability of A plus the probability of B. Students, you have noticed that according to this definition a number a real number lying between 0 and 1 has to be attached to every outcome of the sample space of the random experiment. Yanni for example if it is a biased coin not a fair coin but a biased coin the two possible outcomes are head and tail and according to this definition some number between 0 and 1 will be attached to the outcome head and some number between 0 and 1 will be attached to the outcome tail and according to the second and the third axioms the sum of these two numbers should be equal to 1. Now, why did I say according to the second and the third axioms? Yanni second axiom said the probability of the sample space S in other words the probability of the shore event has to be 1. And if we talk about the probability of head or tail the probability of the shore event then according to the third axiom the probability of head union tail where union represents or should be equal to the probability of head plus the probability of tail. So, Saribhat ka lubeh luwab yeh ke dohi outcomes hai, dono exhaustive hai agar aap unka union lein to puri sample space exhaust ho jatiye. To third axiom ke tahit probability of head or tail has to be probability of head plus probability of tail and according to the second axiom because head and tail are exhaustive. So, this sum has to be equal to 1. Ye itna muskil hai nahi jitna ke aapko pehli defa sunne meh sus ho rahe. I would like to encourage you to study this concept in depth and to reflect on the various points that are the various axioms that I mentioned. And I am sure that if you devote a little bit of time to it you will be able to understand that the three axioms fall in place very nicely and they support each other and we have a proper definition of probability. Or note karne ki baat ye bhi hai ke wo dono definitions jo meh aapko pehli di under the quantifiable definition. Both of them relate to the axiomatic definition. Agar aapka coin fair hai to wo jo number aapka sign keringe head or tail ko they will be half and half. Lekin agar aapka coin biased hai jaya saki meh abhi aapke saad discuss kar rahe thi then you will not be able to assign that number to the outcome head until you first roll that coin a large number of times. If you roll it a large number of times and then determine the proportion of times that you got ahead then that proportion is that number that you will assign to the outcome head. Let us now consider another example. The table that you now see on the screen shows the number of births in England and Wales in 1956 classified by sex and whether it was a live birth or a still birth. As you can see the total number of births in 1956 in England and Wales was 7,16,740 out of which 3,68,490 were male births and 3,48,250 were female births. As far as live birth versus still birth is concerned there were 7,335 live births and 16,405 still births. As you will agree there are four possible events in this double classification. Male live birth, male still birth, female live birth and female still birth. The proportions of births in each of these four categories are 0.5021, 0.0120, 0.4750 and 0.0109. Now a male birth occurs whenever either a male live birth or a male still birth occurs and so the proportion of male birth regardless of whether they are live or still born is equal to the sum of the proportions of these two types of birth. That is probability of the male birth is equal to the probability of male live birth plus the probability of male still birth and this is equal to 0.5021 plus 0.0120 and that is equal to 0.5141. Students, you have noted that in this example but towards the end of the discussion I talked about probabilities. I hope you realize that I did so because the total number of births in this example is large enough for me to apply the relative frequency definition of probability and to regard all those proportions as probabilities. And you will be interested to note that the probability of male birth that we obtained in this example just now is almost the same as what we had in the earlier example. You remember that in the earlier example we said that the proportion of male birth for the entire country was 0.514. If we are interested in finding the probability of still birth students, what we should do is similar to what we did just a short while ago. As you now see on the screen, a still birth occurs whenever either a male still birth or a female still birth occurs and so the proportion of still births regardless of sex is equal to the sum of the proportions of these two events. In other words, the probability of a still birth is equal to the probability of a male still birth plus the probability of a female still birth and that is equal to 0.0120 plus 0.0109 and that is 0.0229. So, in this problem we have noticed that we have found on the basis of real data that the probability of still birth in that part of the world in that particular year was 2.29 percent. Students, let us now proceed to the basic laws of probability and these are the laws that will enable you to solve a number of probability problems in a convenient manner. The first one is the law of complementation and as you now see on the screen, if a bar is the complement of an event A relative to the sample space S, then the probability of a bar is equal to 1 minus the probability of A. Let us apply this law to a simple example. Suppose that we toss a fair coin four times and we are interested in finding the probability that at least one head appears. As you now see on the screen, the sample space for this experiment consists of 2 raised to 4 that is 16 sample points. The reason being that each of the four tosses can result in two outcomes. According to the multiplication theorem that I discussed with you in an earlier lecture, 2 into 2 into 2 into 2 that is 2 raised to 4 that is 16 ways of tossing a coin four times. As you will agree, this rule of multiplication enables us to count the sample points very conveniently. Otherwise, if you start listing them, it is not extremely convenient. Of course, you can start with head, head, head, head and then you say head, head, head, tail, head, head, tail, head and so on. But it is possible that after writing 5 or 6, you get a bit confused. Now, the rule of multiplication convinces you that even if you are not able to list them, the total number of such outcomes is 16. Now, the event of my interest is that at least one head appears. You will say alright head, head, tail head, tail, head, head, head again you might get confused. So, if we apply the rule of complementation that I just discussed with you, you will find that it is much simpler. Jo hamara event hai at least one head, iska jo compliment hai that is no head and it is very easy for us to say ke no head wala to ek hi outcome hai and that is tail, tail, tail, tail. So, we apply the rule of complementation that I just presented. The probability of A bar is equal to 1 minus the probability of A. The probability of getting at least one head is equal to 1 minus the probability of getting no head. No head wala outcome, sif ek hai. So, the favorable outcome is only 1. The total number of outcomes is 16 and so the probability of having no head is 1 over 16, m over n, classical definition. And according to the rule of complementation, the probability of A bar is 1 minus the probability of A. So, the probability of getting at least one head is 1 minus 1 over 16 and that is 15 over 16. The next law that I would like to discuss with you is the addition law or the general addition theorem of probability. As you now see on the screen, according to this theorem, if A and B are any two events defined in a sample space S, then the probability of A union B is equal to the probability of A plus the probability of B minus the probability of A intersection B. This equation can be stated as follows. If two events A and B are not mutually exclusive, then the probability that at least one of them occurs is given by the sum of the separate probabilities of the events A and B minus the probability of the joint event A intersection B. We said probability of A union B is equal to probability of A plus probability of B minus probability of A intersection B. This means probability of A union B. Students, do you not remember that A union B means either A occurs or B occurs or both A and B occur. At least one of the two A and B occurs. Or right hand side, the last term probability of A intersection B. Again, do you not remember that A intersection B represents the joint event that both A and B occur? The way I presented the theorem just now, that is perfectly correct. I will once again repeat it for you according to the general edition theorem of probability. The probability that either A or B or both of them occur. In other words, the probability that at least one of the two occurs is equal to the probability of the first plus the probability of the second minus the probability that both of them occur. This theorem key application or its key examples, we will discuss in the meantime. I would like you to study all that has been discussed today and to practice with various questions that you are able to. I wish you the very best and until next time, Allah Hafiz.