Blog Posts
Statistics curiosities, applied Mathematics, gaming ethics, and more — from the history of heads or tails to Pascal's triangle, the Galton board, random walks, and the gambler's fallacy. Stay tuned for the latest blog posts.
Written by Paul MeleroRandom Walks
Where coin flips meet deeper math: analyzing paths with Pascal's triangle and Catalan numbers, and why a fair coin can look unfair for so long, stochastic processes and the arcsine law.
View the full series →- Part 1 of 3
Intermediate level
What Do Pascal's Triangle and a Coin Have in Common?
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.
Read More ➡️ - Part 2 of 3
Advanced level
How a Fair Coin Spends Most of Its Life Looking Unfair?
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?
Read More ➡️ - Part 3 of 3
Advanced level
Flipping a Coin Until You Get Pi
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 .
Yes, . 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.
Read More ➡️
Basics
How to Flip a Coin Online — And Why It's Fairer Than a Real Toss
Need a quick, fair decision? Flipping a coin online is the fastest way to get one — no coin, no table, no argument about whether the toss was fair. This guide covers how it works, why it matters, and when to reach for an online coin flip instead of digging through your pockets.
Basics
How we toss a coin in FlipTheCoin.app
The mathematics behind a coin flip is simple: a coin has two sides, heads and tails, and each side has an equal chance of landing face up. But how do we actually toss a coin? In this article, we'll explore the science behind coin flips and how we ensure that each flip is fair and random.
Basics
Where does the term "edge case" come from?
Running a coin-flip site, I think about edges a lot.
There's the literal edge of a coin: the thin rim between heads and tails. A real coin landing on that rim is a rare event, roughly 1 in 6000 flips, and yes, our simulation handles it too. And then there's the software engineering "edge case": the term developers use for weird, extreme, boundary-pushing inputs that everyday testing misses.
Are the two related?
Short answer: no. The "edge" in "edge case" has nothing to do with coins, rims, or thin bits of metal. But the fact that a coin-flip app has to handle both — a literal edge and an edge case — is a coincidence I can't let go of. So here's the story.
Written by Paul MeleroProbability Basics
Start here — a guided path from what probability means to how distributions describe a coin's behavior.
View the full series →- Part 1 of 3
Basics
Probability Basics
Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is a way of quantifying uncertainty and measuring the likelihood of different outcomes. In the context of coin flips, probability is used to determine the likelihood of the coin landing on heads or tails.
This article is an introduction to the topic of probability. We will stick to plain language and use a coin as the only example. More advanced statistics will come in future articles.
Read More ➡️ - Part 2 of 3
Basics
Heads or Tails — The Complete Guide to the Coin Toss
"Heads or tails?" Three words that have settled disputes, started games, and made decisions for thousands of years. But where does that phrase actually come from? And is the coin really fair?
Read More ➡️ - Part 3 of 3
Basics
What Is a Probability Distribution?
Flip a coin once and the outcome is a coin toss. Flip it a hundred times and something more interesting appears: a shape. Count how often each result happens and you are no longer looking at luck — you are looking at a probability distribution. This short article explains what that means, introduces the most famous distribution born from coin flips, and lets you watch one build itself, ball by ball, in an interactive Galton board.
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Intermediate level
The Unexpected Statistics of Coin Flips
You have probably flipped a coin at some point in your life. Maybe you flipped a coin to decide who would go first in a game, or maybe you flipped a coin to decide where to go for dinner. But have you ever stopped to think about the statistics of coin flips?