Part A: JEE Main Level

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. In a class, sixty per cent of the students are boys. Thirty per cent of the boys wear glasses, and twenty per cent of the girls do. Now a student with glasses is chosen. What is the chance that this student is a girl? One friend says forty per cent, because forty per cent of the class are girls. Another says twenty per cent. So who is right? Keep your answer ready; Example two settles it. This is Part A of our unit problem bank, at JEE Main level, and quick questions like these come up often. So, let's begin.

0:39Part A: a 30-minute test

So, here is what Part A holds. Sixteen solved problems, all at JEE Main level. Short, quick questions, okay? And they are completely mixed: the mean, the variance, counting, independent events, Bayes' theorem and the binomial, all together. Nobody tells you which chapter a question comes from, just like the real exam. Right? The plan, from the chapter overview: finish Chapters one to six, then attempt Part A as one thirty minute test. Before that, let us sharpen two tools, and solve two problems from Part A together. With me?

1:17Tool 1: variance without the data

Tool number one, from Chapter one: the variance. Sigma, the Greek letter, is the standard deviation, so sigma squared, with the small two, is the variance. The big capital sigma means the sum of, the small i written low is a subscript, and x bar, with the bar on top, is the mean. So sigma squared equals the sum of x i squared, over n, minus x bar squared: the mean of the squares minus the square of the mean. And we divide by n, as our course does. Okay? Now, many questions give no data at all, only n, the mean and the variance. Can you still find the sum of the values, and the sum of their squares? Yes, you can, and it is really simple. The mean is the sum over n, so the sum of x i equals n times x bar. Right? And for the squares, just turn the first line around. The sum of x i squared equals n times, in brackets, sigma squared plus x bar squared. So here is the topper's habit: when the data are missing, rebuild these two totals first. Got it?

2:26Tool 2: total probability and Bayes

Tool number two, from Chapter four. Look at this tree. From the black dot, chance first picks one case: E one, E two or E three, and the small numbers are subscripts. P of E one, along the branch, is the probability of that case. Then event A happens, or it does not; A prime, with the little dash, means it does not. Okay? The cases E one up to E n, the three dots meaning and so on, form a partition. No two overlap, together they cover the whole sample space, and each has a positive probability. So exactly one of them occurs. Right? Now multiply along each branch that ends in A; those are the products on the right of the tree. The vertical bar is read as given, so P of A given E one is the chance of A once E one has happened. Add the products, using the capital sigma again, and you get P of A, the total. This is the theorem of total probability. Clear? Now let us turn it around. Suppose we know that event A has happened. How likely is it that it came through case number k? That branch's share of the total, yes, exactly! P of E k given A equals P of E k times P of A given E k, over the same sum of all the branches. This is Bayes' theorem: your branch over all the branches that end in A, added up. With me? And always check two conditions. The cases must form a partition, and P of A must be greater than zero, that is the greater-than sign on screen, because it sits at the bottom of the fraction. Okay?

4:02Example 1: sum of squares from mean 6 and variance 4

Alright, here is Example one, an easy one from our set. Three numbers have mean six and variance four. Find the sum of their squares. The numbers themselves are not given. Pause the video and try it first. Done? Okay, let's check. The number of values, n, is three, x bar is six, and sigma squared, the variance, is four. Okay? Now, many students see the four and square it again, because the formula says sigma squared. So they use sixteen. Is that right? No, no, wait, that is the first trap! The question gives the variance, and the variance is already sigma squared; sixteen would be correct only if four were the standard deviation. So we use our rebuilt total: the sum of x i squared equals n times, sigma squared plus x bar squared. Right? Now, x bar squared is six squared, which is thirty six. Four plus thirty six is forty. So the answer is forty, isn't it? Oh no, careful, forty is only the mean of the squares! We still multiply by n, which is three. Three times forty, where the cross means times, gives one hundred and twenty, the course answer. Clear?

5:24Example 2: student with glasses is a girl, P = 4/13

Now, remember our puzzle from the start? Here it is as Example two, a medium one from our set. Sixty per cent of a class are boys; thirty per cent of the boys and twenty per cent of the girls wear glasses. Then a student with glasses is chosen. Find the probability that the student is a girl. Pause the video and try it first. Done? Okay, let's check. The cases are B, a boy, and G, a girl. Every student is exactly one of the two, so they form a partition. P of B is zero point six, and P of G is zero point four. Okay? Let A be the event that the student wears glasses. P of A given B is zero point three, and P of A given G is zero point two. Right? Now, twenty per cent of the girls wear glasses. So the answer is simply zero point two, isn't it? That is not right; it is the classic trap. Zero point two is P of A given G, but we want P of G given A, the other way round. So first, total probability. The boy branch, zero point six times zero point three, gives zero point one eight. The girl branch, zero point four times zero point two, gives zero point zero eight. Add them, and P of A is zero point two six. With me? Now Bayes: the girl branch over the total. Zero point zero eight over zero point two six is eight over twenty six, which simplifies to four over thirteen. And P of A is not zero, so the division is safe. Got it? So, back to our puzzle. Four over thirteen is about zero point three one; the wavy equals sign means approximately. So it is not forty per cent, and not twenty per cent either. Seeing glasses pulls the chance of a girl down, because boys wear glasses more often. Both friends slipped. Clear?

7:24Quick revision: variance shortcut and Bayes on a tree

So, quick revision. First, the variance is the mean of the squares minus the square of the mean, dividing by n. Second, when the data are missing, rebuild the sum of x i squared as n times, sigma squared plus x bar squared, and a given variance is already sigma squared. Third, Bayes' theorem is your branch over the sum of all the branches ending in A, once you have a partition. Now attempt Part A as one thirty minute test, and try the practice problem on this page. All the best!