Random Walks

There is a triangle of numbers that has fascinated mathematicians across the globe for centuries. Its construction is so simple a child could draw it in a notebook. And yet, hidden within its rows lies a deep connection between three branches of mathematics that, at first glance, seem to have nothing in common: combinatorics, algebra, and probability. The most surprising part? That connection reveals itself through something as ordinary as flipping a coin.

Here is a question that sounds simple and isn't. Suppose two friends, Alice and Bob, play a game. A fair coin is tossed once per second. Heads, Alice scores a point; tails, Bob does. They keep playing for a whole year — more than thirty million tosses. At every moment, one of them is "in the lead" (has more points so far). What is more likely?

Matt Parker, from the Stand-up Maths channel, recently published a video with a premise as simple to state as it is hard to believe: if you flip a coin repeatedly until heads outnumber tails, and compute the average ratio of heads to total flips... the result is $\frac{\pi}{4}$.

Yes, $\pi$. The number that belongs to circles. Hiding inside coin flips.

In our previous article on Pascal's triangle we saw how a triangle built from simple addition contained, within its rows, the seeds of combinatorics, algebra, and probability. Today we are going to pull on that same thread. And it will take us somewhere no one expected.