The Tale of Variance

Once upon a time, there was a math teacher. He had a unique intention: not only he wanted his kids to learn the subject, but also understand how mathematics work. The most difficult part was definitely talking about infinity in mathematics and variance. But he found out a genius way to present the latter, and it didn’t even take him infinite time. His method has proven so successful during the years, yet so simple.

He walks in the classroom and starts the lecture with a simple exercise. He gives a coin to the class and draws a line in the middle of the chalkboard. The students’ task is to toss the coin 100 times and write down the results. But before tossing, choose a random half of the board and write down 100 times heads or tails. Then write down the actual results on the other half. If he cannot find out which one is the created one and which one is tossed, the class can leave for the day.

He walks out of the room, drinks a hot fresh coffee, then 10 minutes later comes back.

Then walks in, thinks for a little while, examines the patterns, then he picks the correct sides.100% of the time. Always. He is never wrong.

How can the teacher be always right? Does he cheat? He surely has CTV cameras in the room and runs to the security. But he never sweats. So he must have a special choke where the early parts have different colors, right? The answer is no.

He actually understands variance while the students do not. We, as human beings tend to organize thing in our head. Our brain dictates that if the coin toss will result both heads and tails in 50% in the long run, that should look more like H-T-H-T-H-T-H-T-H-T than HHHHH-TTTTT.

And here comes the important part for serious gamblers: If the last toss was heads, we think the next toss will more probably be tails than heads. But if we have a genuine coin and not one purchased from a magicians’ store, the result of the next toss will be independent from the previous ones and will have equal chances for heads and tails.

Let’s get back to our teacher. How does he find out which line is created and which is tossed? He simply looks for the longer streaks which would seem odd for an amateur eye.