Fundamental Physics-1

Añadido: 2026-05-11

[00:00:00][music] >> Swayamprabha, digital India, educated India.

[00:00:09]>> [music] >> So, in thinking about what I should talk to you about in this lecture, uh there are many topics presented themselves as very suitable.

[00:00:40]But then I decided on one which is used in all subjects, no matter what you're going to do, that has to do with elementary notions of what probability theory.

[00:00:50]The other advantage it has is that probability is one of the most misunderstood subjects of all.

[00:00:56]And this, of course, is immediately apparent in our daily discourse. We find that the people misuse the concept of probability. They make arguments which are fallacious or wrong arguments based on probability and so on.

[00:01:10]So, I'd like to talk to you about probability to start with, and then we go on to a little bit of statistics and probability distributions and so on.

[00:01:17]The applications are plentiful, so I won't name any specific applications except a few cases, but then they're plentiful all around you.

[00:01:26]The mathematics needed for it is at the same time very simple and very sophisticated, depending on the level at which you want to approach it.

[00:01:35]Probability theory itself was put on a firm mathematical foundation, a rigorous mathematical foundation, in the last century, around the middle of the last century, the earlier part of the last century, by people like Kolmogorov and Smoluchowski and so on.

[00:01:52]We're not going to go into the history of it, but I'm going to give you a large number of examples which we can work out. So, we will essentially work out a few problems in probability theory and see how it's applied.

[00:02:03]And you will find it appearing in surprisingly large number of places.

[00:02:08]Okay.

[00:02:09]Here is an argument. Let me start with an argument which is fallacious, which is wrong.

[00:02:14]It's so often used in daily life that we even don't even realize that this argument is being used.

[00:02:21]Take a lawn where blades of grass are growing. There are millions of blades of grass growing on this lawn, assume.

[00:02:28]And I take a cricket ball and throw it on this throw it up, and it lands somewhere on this lawn.

[00:02:33]It hits a blade on which there is an ant and it crushes the ant. It injures the ant.

[00:02:38]And the ant thinks to itself, "This has got to be by design. It's purposeful.

[00:02:44]It's not accidental.

[00:02:45]And the reason the ant says it's purposeful is because it says of the millions of blades of grass on which the cricket ball could have landed, it landed on my blade of grass and injured me.

[00:02:56]Therefore, it should have been done with great premeditation and by design.

[00:03:01]And it's not random at all."

[00:03:04]Would you agree with this argument?

[00:03:07]No, sir.

[00:03:08]What's wrong with it? It's just that Let's suppose there are a million blades of grass and the probability that it lands on any one is one in a million.

[00:03:19]But it happened. One in a million is a small number, 10 to the minus six.

[00:03:23]But it happened. So, the ant says, "It's got to be by design. The probability is too small that it would have hit me, but it hit me. Therefore, it's got to be by design."

[00:03:32]Would you agree with that? No, sir.

[00:03:34]No. Why not? He thinks that he thinks that every blade of grass Yeah. and then every ant would say Yeah, exactly. There's an ant on every blade of grass, let's assume. And one of the ants has got to be hit, so this ant got hit.

[00:03:47]And it says it was deliberate by design.

[00:03:50]But we see this argument being given all the time in daily life.

[00:03:53]That this has got to be by design.

[00:03:56]Because it's too intricate and too complex and smaller probability for it to happen randomly. Therefore, it's happened to me. Therefore, it's by design.

[00:04:04]Completely wrong argument because now you're talking about after the event.

[00:04:09]After the ball has landed on some blade of grass, the probability that it would hit that blade of grass is one.

[00:04:15]Because it hit that blade of grass.

[00:04:17]That's the end of it. So, you can't now say the probability was 10 to the minus six. It landed on me and therefore it's got to be by design.

[00:04:25]Now, a lot of things in daily life like astrology and so on are based on arguments like this.

[00:04:32]They're based on purely anecdotal evidence to start with, but they're based on spacious arguments of this kind which are wrong.

[00:04:38]So, we'll try to straighten out some of these things and give a proper definition of probability, etc. And I'll start with the simplest machine of all for probability, tossing a fair coin.

[00:04:50]I toss a coin and it's got two possible outcomes, a head and a tail. And let's use the symbols for it. So, I will denote the head by H and tail by T.

[00:05:02]These are the two possibilities and no other possibilities. We are ignoring the possibility that it lands on its edge and remains balanced, etc. We take it to be zero.

[00:05:11]And the probability of only H and T. If the coin is fair and you make a coin toss, the probability that you get either H or T, the word or is very important, is exactly one half.

[00:05:25]Why do I say that?

[00:05:28]Why do I say the probability is 50/50 that I'm going to get heads or tails before I make the toss?

[00:05:36]So, what?

[00:05:39]Uh-huh.

[00:05:41]We'll do that experiment. We can do it on a computer right right Those of you who can program a machine, do that right away. Toss a fair coin 100 times and find out if you're going to get exactly 50/50. On average, you will get Now, what's meant by this average? The probability Both the events are equally likely.

[00:06:00]So, I want to define the statement. What is meant by the saying that the probability of a head is a half?

[00:06:06]I want to define this probability properly and then show that it's a half.

[00:06:10]That's a separate problem, but I want to first define the probability. There are many ways of doing this, but we want an operational definition which you can actually check out.

[00:06:23]The number of ways in which a desirable event can He says the number of ways in which a desirable event. What's a desirable event here?

[00:06:34]Okay. So, you want the outcome to be what you want, namely the head.

[00:06:39]Let's take head.

[00:06:40]H is the head and I want heads. So, what should I do with this coin? Maybe heads and tails and there are total ways two ways in which you can get outcome. Now, you're saying that if I toss toss the coin twice, I'm going to get one head and one tail. Is that what you're saying?

[00:06:56]You didn't say that. Uh so, what exactly is How do you define this probability?

[00:07:00]When you say I have a 50/50 chance of getting head or tail, how do I define this? You say that it's very unlikely that you will get something that's far away from Like 99.1 We'll we'll we'll come to that. We'll quantify. Eventually, we're going to quantify. What's the probability that you get 90 heads and 10 tails and so on? We'll do that.

[00:07:21]But I want to first, even before that, begin by def- defining probability.

[00:07:26]How do I define probability at all? So, if you if you take a one coin and you flip it like multiple times, how many of the times you want Right. So, then there you go. I conduct the same experiment under identical conditions repeatedly, and then what?

[00:07:41]You count the number of times you get a head and you count the total number of Okay, he's on the right track. We do the following. We keep tossing this coin, identical coin, over and over again.

[00:08:01]Keep track of the number of heads. The rest have to be tails, because there are only two outcomes, right? And then you calculate the following. You calculate number of heads divided by total number of tosses.

[00:08:27]You divide one by the other. And then what? I have I conduct this experiment a million times, so the denominator is 1 million, and the number of heads, say, was 722,462.

[00:08:40]So, I take that ratio, and then what happens?

[00:08:50]But that will depend on the total number of heads tosses, right? I get different numbers, but why don't I just do it in one big experiment?

[00:09:01]As you increase the total number of tosses Yes. the number of heads will approach a specific value.

[00:09:07]>> Right. Total limit. You're guaranteed that it will approach a limit. In other words, if n is the total number of tosses, then this ratio is H over n, but capital H is the total number of heads, number of times I get little h as the outcome, right? And then you're guaranteed that limit n tends to infinity H over n equal to a finite number which is less than one. And that's the a priori probability that you get a head.

[00:09:53]That's the probability before you do the experiment that you take a coin and look at it and say what's the probability that if I toss it I'm going to get a head.

[00:10:01]Okay? This is called the law of large numbers. There's no reason why this limit should exist.

[00:10:07]But there are deep mathematical tests to why it actually does exist, but this is a good operational definition for our purposes of probability. Because after all if there is a limit it means that this number of heads is oscillating about it and then actually tends to it as N tends to infinity.

[00:10:26]There's no reason why a sequence like this should converge at all. It could just be up and down flipping back and forth.

[00:10:32]But you're guaranteed by what's called the law of large numbers that a limit exists and is the a priori probability.

[00:10:39]A priori before the fact probability getting a head.

[00:10:43]Right?

[00:10:44]Now you can easily generalize this to the situation where the coin is a biased coin.

[00:10:49]Suppose the coin is heavier on one side than on the other and lands on heads more often than it does on tails.

[00:10:55]Then there is a probability P which I'll denote by little P. There's an a priori probability P which is a number between zero and one that you get a head on a single given toss.

[00:11:13]Any given toss.

[00:11:16]If you have only two outcomes in the experiment, P and heads and tails then the probability of getting a tail must be one minus the probability of heads. Because the total probability of some outcome is one.

[00:11:30]Some outcome or the other is always one.

[00:11:32]So, we will say P is the problem Oh, sorry.

[00:11:42]of H and Q equal to 1 minus P equal to 1/2 of T.

[00:11:56]If P is equal to Q is equal to 1/2, what do you call the coin?

[00:12:01]It's a fair coin. Otherwise?

[00:12:04]a biased coin, okay?

[00:12:06]Now, a process of this kind where you have only two possible outcomes and the probabilities of the two is equal to one is called a Bernoulli process. So, if I'm tossing this coin, what I'm doing is to demonstrate of a Bernoulli process.

[00:12:22]And now I'm going to do the experiment.

[00:12:24]Take a coin, and we could as well take a biased coin to start with because we can solve a more general problem. A fair coin that's gone. I've got a little P as a probability of a head. Yeah. So, P here is one and N here is >> P is not one.

[00:12:39]P is equal to P is a number between zero and one.

[00:12:42]That's the arbitrary probability that I get a head in a toss. Okay, why do I say the probability for a fair coin? Let's look at the fair coin first. Why do I say the probability of four heads in a row is 1 over 16?

[00:12:54]>> [clears throat] >> No, all the outcomes are equal No, all the outcomes Yeah, every one of these 16 is equally likely. I understand.

[00:13:03]So, if you take the probability of each outcome as P Yeah.

[00:13:07]>> then all of them are equally likely, so all of them must be P. But then the probability of something happening is one, so 16 P For for four heads in a row, what do I need? The first coin turned out to be head.

[00:13:17]What do I need next for four heads?

[00:13:19]Another Another head. It should And so, the first coin should be head, the second coin should be head, and the third coin should be head, and the fourth coin should be head.

[00:13:29]And the four tosses are independent of each other.

[00:13:31]If they affected each other, that would be different.

[00:13:35]So, the coins are independent of each other. When probabilities of different events are independent, the probability of having both events, this and that, multiplies.

[00:13:47]So, the multiplication symbol is shorthand for the word and in daily language.

[00:13:56]Think about it. Whenever you say and, what you mean is multiplying probabilities.

[00:14:05]You want this and that. You want the probabilities of the two to be multiplied, provided they're independent events.

[00:14:12]If you arrange it magically such that with some magnetic coin or something that if it's head once, it'll be head the next time also, then it's no longer true. This is no longer true. But if they're independent events, you want both events to occur or multiple events to occur one after the other, the probabilities multiply.

[00:14:30]Okay?

[00:14:32]What is the probability that you get head head head and tail?

[00:14:38]So, let's write down these probabilities for a fair coin to start with.

[00:14:45]Probability What's the probability of this?

[00:14:56]I specified the order. I've written the word for you.

[00:15:00]Once Once I've written the word for you, the matter is over, right?

[00:15:04]It's also 1 over 16.

[00:15:07]Every one of them is 1 over 16.

[00:15:10]Now, I ask, what is the probability of four heads?

[00:15:19]Equal to 1 by 16. What's the probability of two heads two tails?

[00:15:33]Now now I have made Why do you say one over four?

[00:15:39]Let's do an even simpler one. Let's always take the simplest example. Just two coins.

[00:15:44]What are the possible outcomes?

[00:15:48]Head head, head tail, tail head, tail tail. So now in the case of two heads two coins, what's the probability that both are heads?

[00:15:57]Because it's a product of half times half. And what's the probability of a head and a tail?

[00:16:04]Why do I say that?

[00:16:07]Because while each event has got probability 1/4 to get a head and a tail, I don't tell you which one comes first.

[00:16:16]So you got one head and one tail and you can either realize it by HT or TH.

[00:16:21]So either this or that. And that tells you then you must add the probabilities.

[00:16:26]So the word and implies multiply probabilities for independent events.

[00:16:59]Yeah, for two H and two T, you have to choose two heads out of four coins.

[00:17:05]The number of ways of choosing two objects out of four is 4C2.

[00:17:09]And once you've chosen the heads, the rest are tails.

[00:17:11]So we will write down the explicit probabilities in a minute in the more general case.

[00:17:23]Absolutely.

[00:17:25]Absolutely. Exactly. We're going to generalize that statement in a minute.

[00:17:30]Okay?

[00:17:32]Yeah, that's 4C2. It's exactly the same as a binomial coefficient. So, we're going to write down the general formula, which you can guess.

[00:17:39]Maybe.

[00:17:40]Now, I go back and ask a general question. So, is this completely clear?

[00:17:44]When you have independent events and you want two events to happen, this or that, then the probability is add. If you want this and that, then the probability is multiply.

[00:17:53]And at all times, the probability is the total probability is one.

[00:17:58]It's normalized to unity. If there is no total probability conservation, some event happens which is not included here, then the total probability can be less than one.

[00:18:08]Suppose I give you a fair coin and I ask, "What's the probability of a head or a tail?" And there is a probability 1/8 that the coin falls into the ditch and disappears.

[00:18:18]Then what's the probability of getting a head?

[00:18:27]7/16 because there's only 7/8 probability that the coin survives. And then you have half probability that it Okay, so that's just common sense. Now, I'm going to ask in n coin tosses, the probability in n tosses of h equal to First of all, in n tosses, how many What What is the sample space of h? H is the total number of tails in n tosses. Now, we're going to do an unfair coin with probabilities p and q of heads and tails. We're going to toss it n times.

[00:19:03]We're going to record the results, and I ask, "What is the probability of getting a capital h as the total number of heads?"

[00:19:12]What is the sample space of H? What can H range over?

[00:19:17]First, whenever you talk about the probability of a random variable, you have to specify the sample space.

[00:19:23]What are the possible sets of values?

[00:19:25]What's the possible set of values of this number?

[00:19:29]It could be zero.

[00:19:33]It could be one. It could be two.

[00:19:35]It could be all n.

[00:19:39]n plus one possibilities.

[00:19:54]What's P n of H equal?

[00:19:56]First of all, you're supposed to get capital H heads.

[00:20:00]The remaining must be tails.

[00:20:03]Right? So, what does this imply?

[00:20:06]The total number of heads plus the total number of tails must be equal to n, which is a given number, the total number of coins at your disposal.

[00:20:16]So, it's sufficient to work with capital H as the variable because for a given capital n, T is just n minus H.