The Law of Large Numbers for Kids: Why Coin Tosses Even Out

0:00 Welcome to Mass Is Not Boring by Kidopoly.com. I'm Mira. And I'm Finn. If you haven't already, tap follow wherever you're listening so you never miss an episode.

0:10 Okay, Mira, what are we exploring today? You've got a shiny coin sitting right in front of you.

0:17 I do. Finn, close your eyes for a second. Imagine I flick this coin high into the air. It flips end over end, flashing in the light, and slaps down onto the back of my hand. Can you tell me, with absolute certainty, if it landed on heads or tails?

0:36 No, nobody can do that. It's completely random. It's an even chance, 50-50.

0:43 Exactly! One single toss is completely wild. Literally anything can happen. But here is the strange, almost impossible secret of mathematics. If you toss it just once, it's wild. But if you toss it 10,000 times, something completely steady and predictable magically appears.

1:05 Wait, what? How can doing something wild 10,000 times make it steady? The coin doesn't know how many times it's been tossed.

1:14 No, it doesn't. And that is the mystery. It is a mystery that has puzzled the brightest minds for hundreds of years. Today, we are going to look at the invisible rule that turns chaos into order. To really see this idea, we need to travel back to the Second World War. The year is 1940 and we are in Denmark.

1:37 There is a mathematician from South Africa named John Kerrick.

1:42 A South African in Denmark? What was he doing all the way up there? He was just visiting family when the country was invaded.

1:50 He couldn't leave, so he was held in an internment camp called Hald. Now it was a gentle place for him, but he was stuck there. He had hours and hours of empty time.

2:03 That sounds incredibly boring. Just waiting for the war to end.

2:08 Right, but John Kerrick was a mathematician. And mathematicians don't just see boredom, they see an opportunity to run an experiment. He had a fellow internee there named Eric Christensen.

2:22 Let me guess.

2:23 They played a game? The longest game of chance you can imagine. They found a coin, and they decided they were going to toss it 10,000 times and write down every single result on paper.

2:37 10,000 times? Their thumbs must have been so sore. Just sitting at a table. Flick, catch, write. Flick, catch, write. For days.

2:49 Yes, it took immense patience. But they weren't doing it just to pass the time. John Kerrick wanted to physically see a massive mathematical truth with his own eyes. An idea that has a very famous name, the law of large numbers.

3:06 The law of large numbers. That sounds grand, like a giant rule book for the universe.

3:12 It really is. So, let's picture them. John and Eric, filling up pages and pages with tiny tally marks. Heads, tails, tails, heads, heads, heads.

3:26 Okay, but hold on. I think I know how this ends. If the chance of getting heads is exactly even, then after 10,000 tosses, they must have gotten exactly 5,000 heads and 5,000 tails. See?

3:41 That is exactly what almost everyone thinks. But that's not what happened. After 10,000 tosses, John Keritch counted up all his tally marks. He had exactly 5,067 heads.

3:55 Wait, 5,067? But that's not exactly half. Half would be 5,000 flat. Did the law of large numbers break?

4:06 No, it worked perfectly! The law of large numbers doesn't say that you will get perfectly even results. It says that the more times you do something, the closer and closer the overall picture settles down to what you'd expect.

4:21 So 5,067 out of 10,000 is incredibly close to half. It's a tiny, tiny wobble away from perfect.

4:30 Exactly. In the first 10 tosses, they might have gotten seven heads and three tails, which feels wild. But stretch that out to 10,000, and the wildness flattens out. It settles down.

4:45 Okay, but wait. Let me ask you a real question about this. If I'm tossing a coin and I get heads five times in a row, the next one absolutely has to be tails, right? Because it's due. The law says it has to balance out.

5:01 Oh Finn, I am so glad you said that. Because every mathematics professor in the world would jump out of their chair right now. What you just said is called the gambler's myth.

5:12 The gambler's myth? But why is it a myth? You just said the numbers balance out in the end. Because a coin is just a little piece of metal.

5:21 It has no brain. It has no memory. If you toss five heads in a row, the coin does not know that. The very next toss is still exactly an even chance.

5:33 Whoa, that feels so strange. My brain totally wants to believe the tails are hiding in the coin, waiting for their turn.

5:41 It is incredibly hard to wrap our heads around, but that's the beauty of it. The law of large numbers doesn't force the very next toss to be anything. It just tells us that over a massive amount of time, those tiny, unpredictable moments form a smooth path.

5:59 So who actually figured this out first? Did someone have to toss a coin a million times before John Kerritch did?

6:06 No. The person who proved why this works did it entirely with mathematics. His name was Jacob Bernoulli. He lived in Switzerland in the early 1700s. He spent 20 years trying to prove this one magnificent law. 20 years? Just thinking about chance and luck? Yes. He puts all his proofs into a huge book called Ars Conjectandi, which translates to the art of conjecturing, or the art of guessing.

6:36 But sadly, Jacob Bernoulli died in 1705 before he could share it with the world.

6:43 Oh no! Did his work just get lost in a dusty drawer?

6:47 Not at all. His family knew how important it was. Eight years later, in 1713, the book was finally published. And when it came out, it completely changed how humans see the world.

7:00 How did it change how we see the world? We don't spend our days tossing coins 10,000 times like John Kerritch.

7:08 Oh, but we use Jacob Bernoulli's math every single day. Finn, this sideways connection is my favourite part of the story. Have you ever wondered how surveys can guess what a whole country thinks?

7:21 I guess they just ask a lot of people questions?

7:24 Right, but human choices are totally wild. You cannot predict what one single person will choose, or what they will decide to do. One person's choice is like one coin toss.

7:36 Wild! Wait, but thousands of answers make a survey.

7:41 Yes! When you have huge numbers of answers, the law of large numbers takes over. The wildness settles down. A researcher can't track one person, but because of Benoli's math, they can predict very closely what the whole crowd will do.

7:56 That is unbelievable. So the chaos actually has rules hiding inside it.

8:02 It does. That is why the math works so well. Ask one person what they think, and it's unpredictable. Ask 10,000, and a clear pattern appears. Benoulli showed us that if you step back far enough, the universe makes sense.

8:17 That makes me look at ordinary things totally differently. Even just dropping a handful of marbles on the floor. It's not just a mess, it's a giant mathematical law at play. Alright, Finn, quiz time. Let's see what you remember about the law of large numbers. Bring it on, I am totally ready.

8:36 First question. What is the name of the mathematician from South Africa who tossed the coin in the camp? Next one. How many times in total did he and his friend toss that coin? Question three. How many heads did they actually get? Here's number four. What is the name of the mathematician from Switzerland who proved this law?

9:14 Last question. In what year was his famous book finally published? Great job! Let's see how you did. The first answer is John Kerrich. For question two, they tossed the coin exactly 10,000 times. The answer to question three is they got 5,067 heads. For question four, the mathematician was Jacob Bernoulli.

9:45 And the last answer is, the book was published in 1713.

9:50 You know, Mira, I don't think I'm ever going to look at a coin toss the same way again. It's wild that a little piece of metal has such a massive secret hiding inside it.

10:00 I feel exactly the same way. The fact that wildness can settle into something so beautiful just makes the world feel like a magical place.

10:10 And if you loved exploring this idea with us, please leave us a five-star review. Tell us what you thought about John Kerritch and his 10,000 tosses.

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10:42 See you next time on Maths Is Not Boring. Bye everyone.