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Learn Probability Theory from Scratch | Full Course — Transcript · YTtoTranscript

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Learn Probability Theory from Scratch | Full Course

numiqo · 6,896 words · 34 min read · EN-ORIG

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Below is the complete, readable transcript of Learn Probability Theory from Scratch | Full Course by numiqo on YouTube. Read the full text, copy any part you need, or generate a transcript for any video with our free tool.

Hi, and welcome to this complete tutorial on probability theory. In this video, we'll go through the key ideas step-by-step. We'll start with the basics of probability theory, then move on to conditional probability, Bayes' theorem, and finally, the central limit theorem. So, in this video, I've brought four separate videos together in the right

order to give you a complete picture. If you like to explore the topic in more detail, you can also find the content in our book on probability theory. So, let's start with the first question. What is probability theory? Probability is just a way to measure how likely something is to happen. That's it. If

something is impossible, its probability is zero. If something is guaranteed, its probability is one. And everything else is somewhere in between. For example, imagine a normal six-sided die with numbers one through six. What's the probability of rolling a seven? It's zero, because it's impossible. What's the probability of rolling any number from one to six? It's one, because it's

guaranteed. And the probability of rolling a specific number, for example, a three, is somewhere between zero and one. So, probability is really about answering one question. How likely is an outcome? To answer that, we first need three basic concepts. Outcome, event, and sample space. And just a quick note, if you like, you

can also read about this in our book, probability theory. So, let's get started. So, what is an outcome? That's simple. An outcome is one possible result. If you flip a coin, one possible outcome is heads. The other possible outcome is tails. If you roll a die, one possible outcome is one, and the other possible outcomes are two,

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three, four, five, and six. On a roulette table in a casino, when the wheel spins, one possible outcome is the ball landing on zero. But, of course, it could also land on any number up to 36. Each of these numbers is an outcome, one possible result of the spin. Now, we know what an outcome is, just

one possible result. But, what is an event? While an outcome is one possible result, an event groups together one or more outcomes that we're interested in. For example, suppose we are playing a game in which we win if we roll either a three or a four. Then, we can write the event we are interested in as A is three

and four. And don't worry about the curly braces. We use them to group things together. Here, we show that event A includes the outcomes three and four. So, and now we want to know the probability of that event. We simply write this as P of A. Here, P stands for probability, and A stands for the event.

So, P of A means the probability of event A. Okay, but how do we calculate probability? For that, we need a sample space. What is a sample space? The sample space is simply the list of all possible outcomes. So, if you flip a coin, the only possible outcomes are heads and tails. So, the sample space is

heads and tails. Omega is the sample space, and just like events, we write the sample space in curly braces. And for a die roll, the sample space is one, two, three, four, five, and six, since those are all possible outcomes. So, now we have the basic ingredients, possible outcomes, events, and the sample space.

That means we are ready to ask, how likely is an event? If all outcomes are equally likely, then the probability of an event is the number of outcomes in the event divided by the total number of outcomes. So, if you want the probability of rolling a four, that is one possible outcome out of six possible

outcomes. So, the probability is 1/6. Or, if you want the probability of rolling a four or a five, that's two favorable outcomes out of six possible outcomes, so the probability is two out of six. And in general terms, we can write it like this. Here, A is our event, for example, rolling a four or a

five. P of A tells us the probability of that event. So, it tells us how likely it is to roll a four or a five. The bars around A represent the number of outcomes in the event. In this case, there are two. And the bars around omega represent the number of outcomes in the

sample space, which is six here. So, the probability of rolling a four or a five can be written like this. Here, we have the event with the two outcomes, and there we have the sample space. Therefore, we get two out of six. Okay, great. Now, we know how to calculate probability if all outcomes

are equally likely. Now, there are different ways probabilities can be written. For example, the probability of rolling a three out of six possible outcomes is 1/6, so as a fraction, or about 0.16 as a decimal, or about 16% as a percentage. These all mean the same thing. Fractions are often easier for calculating. Percentages are often

easier to understand. So, if someone says there's a 20% chance of something happening, that means probability is 0.2. Now, let's talk about something especially casinos love, independent events. When we flip one coin, the probability of getting heads is 50%. But, when we flip it a second time, what is the probability of getting heads? It is

still 50%. So, when you flip a coin twice, the second flip does not care what happened on the first flip. So, if the first flip was heads, the next flip is still 50/50. That's called independence. So, two events are independent if one event does not change the probability of the other. Examples of independent events are coin

flips, die rolls, and roulette spins. And this is important, because many people think, "I got heads three times in a row, so tails must come next, or at least be more likely." But, that's not true. The coin has no memory. Even after three heads in a row, the probability of getting another head is still 50%,

because the coin has no memory. It does not care about the previous results. You might think it's clear that this coin does not know what happened in the three flips before. But, for many people, it's not that obvious. This mistake even has a name, the gambler's fallacy. The gambler's fallacy is the false

belief that after a streak of one outcome, the opposite outcome is now more likely, even though the events are independent. And casinos often reinforce this illusion. To keep players engaged, many casinos have a monitor above the roulette table that shows the past winning numbers. It may show you so-called hot numbers, so numbers that

appeared often in the recent spins. It may also show cold numbers, numbers that appeared less often. Sometimes, it even shows the percentage of odds and even numbers, or red and black. All of this makes it feel as if there is a pattern, as if the game is giving you clues, as if you can study the screen, outsmart

the table, and predict what comes next. But, in reality, this information tells you nothing about the next spin. Why? Because each spin is independent of the last one. The wheel also has no memory, and the ball bouncing around does not care what it did before. This whole monitor does not help you predict the next number.

What it really does is create a feeling that you can trick the game, even though the odds have not changed at all. So, remember, if events are independent, what happened before has no influence on what happens next. But, not all events work like that. Sometimes one event changes the probability of another. For

example, in many card games. When that happens, we speak of conditional probability. But before we get to that, however, we need to look at two important ideas. And and or. In probability, and is called the intersection and is written with this symbol, while or is called the union and is written with that symbol. So, if we have two events,

A and B is written like this and A or B is written like that. Okay. Let's have a closer look at and. In simple words, and means that both things happen at the same time. So, both events must be true at the same time. In probability, this is called an intersection. And here's a simple example.

Suppose we roll a die and event A is rolling an even number, so two, four, and six. Event B is rolling a number greater than three, so four, five, and six. Now, A and B means the number must be even and greater than three. So, we only keep the outcomes that satisfy both

conditions, which means four and six. Let's visualize this. Let's say here we have event A with two, four, and six and there we have event B with four, five, and six. The intersection of the two events is four and six. So, we can bring them together and see the intersection here. So, if winning the game means that

both events A and B must occur, then we win only if a four or a six is rolled. Okay, and what about or? In probability, or means that at least one of the events happens. This is called the union of two events. So, if we say A or B, we mean the outcome is in A or in B or in both.

Let's use the same example as before. Now, we ask which outcomes are in A or B. We include everything that belongs to at least one of the two events. So, we get two, four, five, and six. So, remember, and means both events happen. That is the intersection. Or means at least one event happens. That

is the union. Okay, and now the interesting question. Why do casinos make so much money? Casinos are built around games that look random in the short term, but are predictable in the long term. That is the key idea. A player can absolutely win a round. A player can even win many rounds in a

row. But if thousands of players play thousands or millions of rounds, the math starts to dominate. And the math usually favors the casino. This advantage is called the house edge. The house edge is a small mathematical advantage built into the game. And small is important here. Casinos do not need to win big on every

round. They just need a tiny advantage repeated over and over again. Let's look at the simple example, roulette. Imagine a roulette wheel with numbers from zero to 36. That means there are 37 possible outcomes. If your number comes up, you win 35 euros and you also get your 1 euro back. If it does not, you lose your 1 euro. At

first, that sounds fair. But this is where probability comes in. The chance that your exact number appears is only one out of 37 because the wheel also includes zero. So, there are 37 numbers, not 36. However, if you win, the casino pays only 35 euros plus your euro. Now, imagine betting 1 euro on every number

from zero to 36. You would need to bet 37 euros in total. No matter which number comes up, only one of your bets would win and you would receive 36 euros. So, overall, you would still lose 1 euro. So, even though the game may look fair, on average, you lose 1 euro for every 37

euros you bet. So, casinos design the payout so that over many rounds, the average result is slightly in their favor. This example also introduced

bar means given that. In other words, what is the probability that it will rain given that it is already cloudy? And this is called conditional probability. In other words, it's the probability of something happening given that we already know some other piece of information. So, in general terms, we ask, what is the probability of A given

that B has occurred? But how do we calculate this? Let's look at our simple weather example and imagine we repeat a little experiment many times. We look outside, note whether it is cloudy at the moment, and then we check whether it rains within the next hour. So, each time we record the two things, the

weather now, so if it's cloudy or sunny, and second, whether it rains within the next hour. So, for example, the first time it's cloudy and rains. The next time it's sunny and doesn't rain. Now, suppose we do this 100 times. Now, we can summarize the results of these observations in a so-called contingency

table. Let's say that in total we had 20 cases with rain and 80 with no rain. And overall, we had 40 cloudy cases and 60 sunny cases. For example, we can see that there were 16 cloudy cases where it rained within the next hour and 24 cloudy cases where it did not rain.

So, if we ignore whether it's cloudy or sunny and just look at all 100 observations, we can ask, "What's the probability of rain in the next hour?" It's 20 out of 100 because in 20 cases out of 100, it was raining. That's 20%. But now we look out the window and see that it's cloudy. So, how do we

calculate the conditional probability? There are two ways to do it. One straightforward way and one more formal way. Let's start with the straightforward one. We want the probability that it will rain given that it is already cloudy. So, instead of using the whole data set, we only look at a subset of observations where it was

cloudy. In total, we had 40 cloudy cases and in 16 of these cloudy cases, it rained within the next hour. So, the probability is 16 out of 40 because out of the 40 cloudy cases, it rained within the next hour in 16 of them. Pretty easy, right? If you want a probability of rain when it's cloudy, we just filter

our data to the cloudy cases and use this subset. Now, let's look at the second method, the general formula for conditional probability. We'll go through it step by step. In our case, A is the event it rains within the next hour and B is the event it is cloudy right now. So far, nothing new, but what

about this term here? This is the probability that it rains within the next hour and it's cloudy now. In other words, the probability that both events happen at the same time and we can easily calculate it. We just take the number of cases where it rains and it's cloudy, that's 16, and divide by the

total number of cases, which is 100. So, we get 16%. And PB is also easy. It's just the probability that the weather is cloudy. So, in 40 out of 100 cases, it was cloudy. That's 40%. So, P of rain given cloudy is 16 divided by 40%, which is 40%. All right, before we get to one

last important point, let's make this more hands-on. We'll jump into our interactive web lab. You can find the link in the video description. Here we can see the probability that a ball passes through the green bar and there we can see the probability that a ball passes through the orange bar and here we can see the probability that a

ball passes through both the green and the orange bar. The balls fall randomly and every horizontal position is equally likely. Now, let's see what happens when we change the size of the bars or when we move them left and right. If we only move a bar left or right, the probability that a ball passes through

it doesn't change because its width stays the same. But if we make the bar wider, it becomes more likely that a ball will pass through it. And in the extreme case, if the bar covers the full width, the probability is 100%. But what we're really interested in is the conditional probability shown down

here. This is the probability that a ball also passes the orange bar given that it already passed the green bar. So, we can think of it like this. If the ball passes the green bar, what is the probability that it also passes the orange bar? So, we take the probability that the ball passes through both bars,

so this area, and divide it by the probability that it passes through the green bar. So, basically, the conditional probability depends only on the size of this green area and the size of the overlap. If we increase the width on this side, the conditional probability doesn't change because the overlap between the two bars stays the same. But if we

expand it on the other side where the overlap changes, then the conditional probability changes, too. And of course, if we change this area, the conditional probability also changes. For example, if we make the green bar very wide but keep the overlap with the orange bar small, then the conditional probability becomes small. That's because many balls

pass through the green bar, but only a few of those also pass through the orange bar. So, if the green bar is very wide compared to the intersection, then knowing a ball pass through the green bar doesn't give you much information. But in this setup, if a ball passes through the green bar, it's quite likely

that it will also pass through the orange one. In the extreme case, if the orange bar completely covers the green bar, then every ball that passes through green will automatically pass through orange as well. And then, the probability that a ball passes the orange bar given that it passed the green bar is 100%.

And of course, we can also move the green bar completely outside the orange bar. Then it becomes impossible for a ball that passed through the green bar to also pass through the orange bar. So, the conditional probability drops to zero. Now, one point to consider. In the last video, we talked about the

basics of probability theory and independent events like coin flips or dice rolls. Of course, you can also calculate conditional probabilities in that case. For example, when flipping coins, we can ask, "What is the probability of getting heads given that we got heads on the previous flip?" But since the flips are independent, that

information doesn't help. The probability is still 0.5. So, for independent events, the conditional probability P of A given B is equal to the probability of A since B does not give us any extra information. So, in my gambling games like rolling dice, flipping coins, or spinning a roulette wheel, conditional probability usually isn't that important because

each round is independent. But in other games, conditional probability really matters, especially in card games. For example, take a full deck of 52 cards. In a full deck, there are four aces, so the probability of drawing an ace is four out of 52. But now imagine one card has already been removed and I tell you

it was an ace. Now, the situation has changed. There are only 51 cards left in the deck and only three aces left. So, the probability of drawing an ace is now three out of 51. So, we gained new information and the probability of drawing an ace changed. And this is exactly why card games feel different

from coin flips or roulette spins. In coin flips and roulette, we usually treat each round as independent. But in card games, once cards are removed, the odds can change from one turn to the next. Okay, one last point. What if you know P of A given B, but you actually need P of B given A? That

probability flip is what Bayes' theorem is all about. Now that we've discussed the basics of probability theory and conditional probability, we move on to the Bayes' theorem. What's the most important theorem in statistics? I would say it's Bayes' theorem. It's behind medical tests, spam filters, and even machine learning. And Bayes' theorem is easy. It's all about

prior knowledge. So, given that you watched my last video on conditional probability, it's quite likely that you'll easily understand this video. Phew, I spent way too long on that line. Let's get started. Bayes' theorem is based on conditional probability. So, a quick reminder, what is conditional probability? In conditional probability, we ask for

the probability of event A given that event B has already occurred. But that we already know. So, what does Bayes' theorem actually do? Let's look at an example. Imagine you work in a hospital and you develop a medical test to find out whether someone has a disease. Now suppose this disease is very rare and

the test is quite good but not perfect. So sometimes it gives you the wrong result. As the doctor, you mainly interested in the probability that the test is positive given that the person has the disease. So the question is, if a person is diseased, how likely is it that the test is positive? Okay, but now let's

switch roles. You no longer the doctor, you're the patient who just tested positive. And now the question changes completely. Of course, you don't care about the probability of a positive test given that you have the disease because in your case, the positive test result is already fixed. You tested positive. But you know that the test is not perfect.

So what you really want to know is not this but that. You want to know if the test is positive, how likely is it that you actually have the disease? So you don't care about the probability of a positive test given that you have the disease because in your case, the positive test result is already fixed.

You tested positive. So you want to know if you tested positive, how likely is it that you really have the disease? So the doctor asks one conditional probability question but the patient cares about the reverse one. And Bayes' theorem is exactly the tool that connects the two. So let's look at how this works.

This is the formula for calculating the probability of A given B. We do that by taking the probability of A and B and dividing it by the probability of B. So this is the conditional probability we discussed in the previous video. Now this is the formula for calculating the probability of B given A. So we switched

A and B. And now let's cut the parts of the formula into pieces. Basically, here it doesn't matter whether we write the probability of B and A or the probability of A and B. It is the same. So let's just switch A and B so we can see that this is the same in both cases.

Now of course, we can also write it in this form. And now we can just plug P of A and B into the equation above. And we have a relationship between P of A given B and P of B given A. So this is our final formula. If you know one conditional probability as well

as the probability of A and the probability of B, you can work out the other conditional probability. Now you might ask, why not just calculate it directly? In practice, one of these probabilities is often much easier to find while the other is much harder to get directly. And the Bayes' theorem gives you a clear

way around that. For example, in our medical test example and in many medical studies in general, it is often much easier to estimate P of positive given disease than P of disease given positive. Why? Because P of positive given disease can be measured directly in a study. You simply take a group of

people who already know to have the disease, apply the test, and then see how often the result is positive. And that is exactly why Bayes' theorem is so useful. It helps you calculate the probability we care about from the probability that is easier to measure. But now let's look at the broader intuition which makes the theorem really

powerful. We start with prior knowledge, get new evidence, and update our belief. So how does this work? Let's look at a simple example. Anna is creative, sensitive, and loves spending time in museums. Is she more likely to be an artist or an accountant? At first, you might say, "Okay, that definitely sounds more like an artist." But our intuition

can fool us here. To understand what's really going on, let's jump into our interactive lab. So the question is, is Anna more likely to be an artist or an accountant? Here's what we know about her. Anna is creative, sensitive, and loves spending time in museums. And here we see the probability that Anna is an artist given that

description. And now we can play around with the probabilities we need. We can change the probability that an artist fits this description, the probability of being an artist in the first place, and the probability of someone having this description overall. And here we can see the result visually. So let's just play around and say the

probability that an artist fits this description is 66%, the probability of being an artist is 24%, and the probability of this description overall is 40%. So we assume that this description fits an artist quite well. If someone is an artist, they are more likely than the average person to be creative, sensitive, and to enjoy spending time in

museums. So let's go back to Anna. She is creative, sensitive, and loves spending time in museums. Is she more likely to be an artist or an accountant? At first, you might say, "Artist." But here comes the important point. There is a second term we also need consider. How likely it is for someone to be an

artist at all and how likely it is for someone to be an accountant. And in general, there are far fewer artists than accountants. So let's say the probability of being an artist is 24%. Now the question is, what is the probability that Anna is an artist given that she fits this description? Uh the

answer is only 38%. So even though the description sounds like it clearly points to an artist, the actual probability is still below 50%. But how is that possible? Let's make it visual. Imagine 100 people. 24 of them are artists and 76 are accountants. If 66% of artists fit the description, that means about 16 artists

match it. But because there are so many more accountants overall, even a smaller matching rate can still produce more people. In this case, about 25 accountants also fit the description. So the description may sound artistic but the larger number of accountants changes the final result. But we can also look at it from another

angle. Before we even hear Anna's description, we already start with some prior knowledge. We know that 24% are artists and 76% are accountants. So our starting point would actually favor accountant simply because accountants are much more common. Then we get new evidence. Anna is creative, sensitive, and loves museums. That sounds much more like an artist. So

our belief shifts in that direction. And that is the key idea. Only by combining prior knowledge with the new evidence can we make a good estimate. If you'd like to explore the topic in more detail, you can also find the content in our book on probability theory. Now that we've covered the basics of probability theory and

conditional probability and Bayes' theorem, let's move on to the central limit theorem. Ever notice how the universe is widely obsessed with bell curves? If you measure the heights of people, bell curve. Measurement errors, bell curve. Test scores, also some kind of bell curve. So in this video, we answer the question, why is the universe obsessed

with bell curves? And I'll explain the central limit theorem to you. Okay, but first let's answer the question, what is this bell curve? Imagine we measure the height of a huge group of people. Then we just draw a horizontal line and place different sections or bins along it. For example, in this section we put everyone with a

height between 155 and 160 cm. In the next section we put everyone with a height between 160 and 165 cm. Now we take each person's height and put it into the right section. Each time a section gets a person, its bar gets a little taller. After many people, the bars form a clear shape. When you

look at the real world, you can see that most people are around an average height. So this height is the most common. That's why the graph is highest in the center. As you move towards the edges and look at the very small or very large heights, you can see that the bars are much

lower. This is because those measurements are rare. So there are fewer people who are very short and there are fewer people who are very tall. So the right side is also lower. If you then trace a smooth line over the tops of these bars, you see that a mountain or bell curve emerges. So if you draw a graph of how

many people have each height, you often get a bell shape, also called normal distribution. Okay, and now back to the main question. Why are so many things normally distributed? Let's start with something easy, but it's basically the beginning of the story. If I take random stuff like this die, I can roll it multiple times. And again,

instead of only writing down the numbers, I can put them into a histogram. So every time I roll the die, I add one small mark to the number I got. So if I repeat this many times, the small marks stack up and form a bar. Now we have a histogram or a frequency

bar chart that counts how often each number appears. But hey, it doesn't look like a bell curve. It looks more like a flat rectangle. The outcomes are one through six. Each one is equally likely. So the result won't be a bell curve. It will be a flat rectangle. So for example, the probability of rolling a

one is the same as rolling a three. Every number has the same chance to appear. So we get a flat distribution. Okay, but we want a bell curve. So what happens if we don't use one die, but two dice? If we take two dice and roll them again and again, we can add the eyes and record the sum.

Now something interesting happens. With one die, the distribution is a flat rectangle, but with two dice, it starts to look like a triangle. But why? When we sum up the eyes of two dice, we can get values from two to 12. If we roll two ones, we get two. If we roll two

sixes, we get 12. So the only way to get two is with 1 + 1. And the only way to get 12 is with 6 + 6. But what about four? We can get four with 1 + 3, 2 + 2, or 3 + 1. And the king of all sums is seven. Seven has the most combinations,

six ways. So when we complete the picture, we can already see a triangle shape. The middle sums like six, seven, and eight are more likely and the edge sums like two and 12 are less likely. That's why the shape becomes a hill, almost like a triangle. So with one dice, the histogram is

basically a flat rectangle. With two dice, it becomes a hill, kind of like a triangle. But where's the bell curve? So here's the next step. What if we don't stop at two dice? Let's add more dice and always do the same thing. Roll them, add the eyes, and put the sum into a

histogram. And since doing this by hand is a bit boring, let's take a look at our interactive lab. If you want, you can find the link in the video description. Here we can choose how many dice we want to roll each time. And there we can start the simulation. For example, if we select one die and

click start simulation, we can see the possible outcomes from one to six below. And if we wait long enough, all outcomes should appear equally often. So the distribution should be a flat rectangle. To speed it up, we can click roll 100 times, so we roll the die 100 times at once. And if we select two dice and sum the

eyes, we should get a distribution that looks like a triangle. Again, let's speed it up and now we can clearly see the triangle shape. Now let's use five dice. Each time we roll the five dice, we add the eyes and record the sum in the chart. Let's again speed it up. Now the shape

is not a sharp triangle anymore. It starts to look rounder. So it already looks quite bell-shaped. So let's try 12 dice. And after a long time, something crazy happens. The histogram starts to look like a real bell curve. Let's speed it up and roll 100 times a few times. And it looks like a bell curve more and

more. So if we scroll a bit down, we see the final result. If we use one die, we get the flat rectangle shape. If we use two dice, we get the triangle shape. And as we add more and more dice, the curve looks more and more like a bell curve, so like a normal

distribution. But now the big question. Why does adding more and more random dice rolls slowly turn into a smooth bell shape? Here's the simple reason. Each die roll is random. But when you roll many dice, you usually get a mix. A few low numbers, a few high numbers, and many in between. So when you add them up, the total

usually ends up near the middle. So there are many, many ways to get a sum near the middle. But on the other side, to get a very small sum, almost all dice must roll low. In the extreme case, only ones. And to get a very large sum, almost all dice must roll high. That is also rare.

When we add more and more dice, this effect becomes stronger and smoother. And that's why the shape turns into a bell curve. Okay, okay, I think we all get it. If you add up lots of dice rolls and repeat the experiment, you start to see a nice bell curve. Now what exactly is the

central limit theorem? To get there, it helps to look at how it was discovered. So who invented the central limit theorem? Well, it wasn't just one person. It was more like a very long group project stretching across nearly 200 years. One of the first people to notice the pattern was Abraham de Moivre. He was a

mathematician who studied probability. His work was strongly influenced by problems in games of chance. He was kind of a gambling math guy. He wanted to understand questions like this. If you flip a coin many times, how many heads do you get? But de Moivre had a problem. When the number of flips gets big, the

calculations get painful. So he looked for a shortcut. And that's when he noticed something important. If you flip a coin many times and count how many heads you get, those counts start to form, tada, a bell curve. So he discovered this. Lots of random yes no events start to form a normal curve. So if you add up lots of

two option results like yes and no, left and right, or zero and one, the totals tend to form a bell curve as well. And if you don't believe me, let's have a look at the so-called Galton board. Here we drop a little ball from the top. At every peg, the ball can bounce either

left or right, and each bounce is a 50/50 choice. 50% of the time the ball will bounce left, 50% of the time it will bounce right. And at the bottom, we note which section the ball falls into. So, let's just drop one ball. The ball bounces left and right, and finally falls in this section.

And each bounce, left or right, is like a coin flip. You can say left is head and right is tails. And now we see the same kind of magic as with the dice. One ball is unpredictable, but when we drop hundreds of thousands of balls, a clear pattern appears. Guess what? They start to pile up into a bell-shaped

curve at the bottom. And it's the same logic as with the dice. So, the golden board is the central limit theorem in action. Many small random steps add up and create one smooth bell-shaped pattern. For a ball to land all the way on the left, it would have to bounce left at almost every peg.

And since every time the chance is 50%, that's really unlikely. The same is true for the far right. It would need right bounces again and again, which is also rare. But around the middle, there are many different paths. The ball can go left sometimes, and the ball can go right sometimes, and still

end up near the center. So, De Moivre was on the right track. It wasn't the full central limit theorem yet, but it was a big first step. So, let's make the next step and move to Laplace. Laplace looked at De Moivre's idea and said, "Cool, but what if it's not only coins? What if it's many kinds

of random things?" Laplace pushed the idea much further and made it more general. When you add many small random things, the final result becomes smooth and bell-shaped. And finally, the mathematicians like Lyapunov and Lindeberg made it more precise and stronger. So, what's the actual central limit theorem? And don't worry, first I'll tell you the definition, and then

I'll explain it visually so that it isn't hard to understand. So, the central limit theorem says, "Even if you have a distribution that is not normal, it could be exponential, binomial, or log-normal, it could be nearly anything, if you repeatedly take random samples from this distribution and compute the mean of the samples,

then the distribution of those means becomes approximately normal as the sample size gets larger." Okay, and now let's make this clearer with an example. The theorem says we can start with almost any base distribution, so here you can choose different kinds of distributions. Let's pick the exponential distribution. It clearly does not look like a bell curve.

Now we take a sample from this distribution, for example, a sample of 20 points. You can see those 20 points here. From these points, we calculate the mean and plot it below. Then we repeat the same process. Take another sample of 20 points, calculate the mean, and add it to the histogram. And we can do this again and again and

again. So, now we are building a histogram, not of the original values, but of the sample means. Let's click simulate. And now you can see that the distribution of the sample means starts to approach a normal distribution. If we increase the number of points in each sample, the bell curve becomes even clearer. And that is basically the

central limit theorem. You can start with almost any distribution, take repeated random samples with a large enough sample size, and the distribution of the sample means will approach a normal distribution. Of course, the central limit theorem is not magic. It does not work perfectly in every situation. For example, if your data points affect each other, for

example, in time series, the result may not look normal. Or if your sample size is small and your original distribution is very skewed, you may need a much larger sample before it looks normal. If you'd like to explore the topic in more detail, you can also find the content in our book on probability

theory. My name is Hannah, and I hope you liked the video.