Lecture 02 : Examples and Course outline

So, we are not picking f, but we are actually picking the length O f in a random way right and this length decides also the length of the chord D right because it actually uniquely defines well it not it will not uniquely define the chord, but it will at least uniquely define the length of D that is what the probability cares about. So, what are the good lengths of O f right clearly from the previous circle inscribed circle argument the good length of O f is half or less. So, this means that probability that the length D is at least the length A B this is equal to the inscribed radius over the circumscribed circle radius which is 1 right. So, that is half because if you take an f which is which you take O f to be bigger than half then you go outside the inscribed circle and then you get smaller chords. So, you have three possibilities of the probability of same event right it is one half or one third or one fourth, one third and one half. So, they differ so much. So, what is the paradox here I mean paradox must be clear, but how do you resolve the paradox what is the where is the mistake in the math. So, the mistake is that or the reason why you are getting the different probabilities is because you are assigning probability in different ways to the sample space ok. Each of these three processes is defining a different probability function that essentially is the resolution. What you have to remember is there might be many ways to define the probability function on subsets of an infinite infinite sample space omega ok. So, you for example, process 1 defined probability on D comma E process 2 defined it on midpoint of D comma E in process 3 defined it on the length O f and as you look at these different maps you ultimately get different probabilities for this event D greater than equal to A B ok. So, that is why we will need some formalism otherwise we get into this. If you only care about physical interpretation then you might get confused in these examples ok. So, in the end we will formally define everything, but this is a good example to keep in mind. So, this also tells you that probability is not merely counting right because in infinite space you cannot really count counting does not make sense, but probability still makes sense and to get a unique answer you have to formalize it properly right. So, here you see the difference between counting and probability. In discrete space you can say that they are equivalent, but not in infinite space probability is doing something else ok. So, now let us shift gears and move to a less abstract more practical example, but still confusing ok. Let us go to example 5 now an example that is practical and often confused ok. It can cause serious confusion not only in an individual, but even in a society and right now it is very relevant. So, let us see this carefully. So, say a COVID-19 RT-PCR test is 80 percent accurate ok. So, you went or I went to a lab which does this test and they say that its accuracy is 80 percent. Now so, I went for this test and unfortunately it came out to be positive ok which you do not want to happen, but say it happened ok. So, I took this test which is which came out to be positive. Now my question is what can you deduce from this from this information ok. So, what is the probability that I have COVID-19 given this information that the test is positive can I deduce with what probability with what confidence with what chance can I say that I have the COVID virus or I do not have the COVID virus right both the things are possible because the test is not perfect. So, can I assign probabilities to this? Now it is very very tempting to say 0.8 the probability that I have COVID is 80 percent, but this is very wrong this is what many people in fact, most people would not realize that this 80 percent accuracy does not mean that this particular specific test that happened makes me COVID positive with 80 percent chance ok. So, calculating this is more complicated. So, this exact answer the formula for the exact answer you will see later in the course, but without going into the formula I will just say that exact answer depends on it requires the knowledge of one how many people in the let us say in the country have COVID-19 virus and second is if the test was done on everybody how many times is it is the answer positive. So, how many times is the test result in general these are different questions and we do not have information for them right we do not know how many people have the virus in the country and we also do not know how many times will the test be positive because the test cannot be done throughout the nation for everybody right the test is only done on a small sample space. So, both these things we do not have. So, in practice they are somehow approximated and once you have the approximation then I can say what is the probability that I have COVID given that the test was positive right, but this information is currently missing from example 5. So, to say more about this this 1 is called. So, 1 is actually equal to what we say true positives plus false negatives right because true positives are those people who have the virus and the result was positive. So, these are the true positives false negatives are those people who have the virus, but the result was negative right. So, that is what number 1 is. What is number 2? So, this is only about test being positive whether the person has virus or not we are not worried about that. So, this is true positives plus false positives right. So, that is the information extra information needed. So, you need true positives you need false negatives and you need false positives these 3 things you need and then you can do a calculation. Now, interestingly if this number 1 is much smaller than number 2 right which means that very few people have actually the virus COVID-19, but on the other hand the test is many times positive in the general population right. So, then it is a bad test that means, that the test is bad. So, then it is a bad test which implies that probability that I have COVID is very small it is much smaller than it is much much smaller than 0.8 ok. So, in case the test is a bad test then the probability of I having COVID is not at all 0.8 it is much smaller it may be even be close to 0 right. So, basically this test does not give you any information except it confuses you because the accuracy was touted as 80 percent right. So, which is the same as saying. So, 1 much much less than 2 is the same as saying that false positives are much more than false negatives right 1 much much less than 2 means that this true positive plus false negative is much much less than true positive plus false positive which is cancelling out it says that false positives are much more than false negatives. So, it is a bad test. So, the taking the test and getting positive does not mean that you have the virus right. So, you can see that these things are quite complicated right. So, when somebody advertises 80 percent accuracy that just means that the test was done on a small sample space and there people who had the virus test became positive with 80 percent chance and people who did not have the virus test was negative with 80 percent chance, but from that when you take the test you cannot deduce anything you need much more information you need information about the whole sample space the general population right. So, this is a very relevant and contemporary example which of course, you must have thought about. So, this is the reason why. So, this makes testing survey and prediction a very complicated affair right. So, it is prone to a misinterpretation especially in social media or even media because part of the things I mean the initial part is correct 80 percent is correct, but then the conclusion you cannot conclude 80 percent because you are these are two different things. So, this is highly prone to misinterpretation in the media because people can be easily made to convince that themselves that this 80 percent is the same as what you want ok. So, those were the examples I hope these examples give you not just remind you of basic probability, but also give you the challenges that lie in applying probability and why we will need formalism why we will need formulas ok. So, lastly something that you might not have seen. So, lastly another aspect of probability is to prove existence usually of some discrete object ok. So, lastly probability is sometimes used to prove the existence of an object ok. So, sometimes it is also used to prove the existence of an object by calculating you calculate probability of an event if it comes out to be positive then it means that object actually exists ok. So, this is a non-constructive proof it may be constructive, but in general these proofs are non-constructive proofs because they only tell you the existence they do not construct that object ok. We will see some examples towards the end of this course. So, one example I can give you immediately. So, this is by the way called probabilistic method ok probabilistic method in especially in combinatorics achieves this. So, an example is. So, let B be a subset of integers and let it have n distinct integers what you want to show is there exists a subset A of B of size quite large more than a third more than n by 3 there exists a large subset that is sum free ok. So, what is sum free? Sum free just means there are no two elements whose sum is a third element. So, no two elements of A sum up to an element in A ok. So, this is what we want to show that for any subset B of integers there is always a sum free subset that is large right this seems to be a difficult to prove statement because there is no hope of a constructive proof because you do not even know B, B can be anything right. So, we will see a proof or you can try to prove this by using probability those are the directions in which this course will move we will formalize concepts we will define them and then we will apply them to many different examples problems some of which may not have to do anything with probability like this example 6 this is not about probability per say right. So, finally, the outline of the course. So, we will do from next time basic concepts from high school the probability questions that you solved those kind of questions then we will look at random variables expectation moments examples of distributions probability distribution examples and some more advanced things which are very important in computer science applications which is conditional probability like this COVID testing example that you can understand rigorously using conditional probability concentration inequalities again concentration inequalities is something very important in computer science concentration basically means that if I know that expectation or average of a random variable is something let us say 10 then what is the probability that in practice you get a number like 20 or a number like 5 when the average is 10 what is the chance of being far away from it far more or far less. So, there are very useful inequalities about that giving you probability upper bounds also called tail inequalities then an idea about the page rank algorithm right for example, which is used in or a variant of which is used in Google search and finally, the probabilistic method and finally, students who are registered inside IIT Kanpur for them grading. So, tentatively what we can do is we can have quizzes assignments and one exam right it is a modular course. So, looking at the timeline we can have three quizzes we can have three assignments we can have one exam. So, quizzes each can be 15 percent. So, that gives you 45 percent weightage assignment we can have 10 percent. So, 30 percent weightage and remaining 25 percent is for the exam. So, based on your performance you will be given a grade any version of plagiarism will be severely punished. So, plagiarism etcetera will be severely punished which obviously, includes cheating from other students.