The Hidden Geometry of Chance: Why Pascal’s Triangle Contains Almost Everything

Pick up a coin. Flip it three times. How many ways can you get exactly two heads?

You might count on your fingers: HHT, HTH, THH. Three ways. Now ask the same question for ten flips and two heads, and suddenly your fingers aren’t enough. You need a pattern — and one of the most beautiful patterns in all of mathematics has been studied across mathematical traditions for many centuries, hiding inside a simple triangle of numbers.

That triangle is Pascal’s Triangle. And once you see what lives inside it, you’ll never look at probability, algebra, or even the stock market quite the same way again.

Building the Triangle

Start with a 1. On the next row, write two 1s. From here, every number in the triangle is the sum of the two numbers directly above it. That’s the only rule.

Row 0:        1
Row 1:       1 1
Row 2:      1 2 1
Row 3:     1 3 3 1
Row 4:    1 4 6 4 1
Row 5:   1 5 10 10 5 1
Row 6:  1 6 15 20 15 6 1

It looks almost too simple. Just addition. But that single rule — each entry is the sum of its two parents — turns out to encode an astonishing amount of mathematical structure. Let’s pull it apart, piece by piece.

The Coin-Flip Connection

Here’s the first miracle. Remember that question about flipping a coin three times and getting exactly two heads? Look at Row 3 of the triangle: 1, 3, 3, 1.

• The 1 on the left counts the ways to get zero heads (TTT). Just one way.
• The first 3 counts the ways to get exactly one head (HTT, THT, TTH). Three ways.
• The second 3 counts the ways to get exactly two heads (HHT, HTH, THH). Three ways.
• The 1 on the right counts the ways to get three heads (HHH). Just one way.

Row 3 of Pascal’s Triangle is the complete answer to every “how many ways” question for three coin flips. Row 10 answers the same question for ten flips. Row 100 answers it for a hundred.

The numbers in Pascal’s Triangle are called binomial coefficients, and they’re written C(n, k) — read “n choose k” — meaning “the number of ways to choose k items from a group of n, where order doesn’t matter.” The entry in Row n, position k (counting from zero) gives you C(n, k).

So C(10, 2) — the number of ways to get exactly 2 heads in 10 flips — is the third entry in Row 10: 45. No counting required.

A Worked Example: The Probability of a Winning Streak

Let’s make this concrete. Suppose you’re playing a game where you flip a fair coin 5 times. Your friend bets you that you’ll get at least 4 heads. Should you take that bet?

Row 5 of the triangle is: 1, 5, 10, 10, 5, 1. The total of all entries is 1 + 5 + 10 + 10 + 5 + 1 = 32, which is just 2⁵ — the total number of possible outcomes for 5 flips.

• Ways to get exactly 4 heads: 5 (the second-to-last entry)
• Ways to get exactly 5 heads: 1 (the last entry)
• Ways to get at least 4 heads: 5 + 1 = 6

So the probability is 6/32 = 18.75%. Your friend is betting on a less-than-one-in-five shot. Don’t take the bet — but now you know exactly how bad those odds are, and you computed it in seconds using a triangle.

The Algebra Hidden in the Rows

Here’s a second miracle, and it’s the reason these numbers are called binomial coefficients. Take the expression (a + b) and raise it to a power:

• (a + b)⁰ = 1
• (a + b)¹ = 1a + 1b
• (a + b)² = 1a² + 2ab + 1b²
• (a + b)³ = 1a³ + 3a²b + 3ab² + 1b³
• (a + b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴

The coefficients — the numbers in front of each term — are exactly the rows of Pascal’s Triangle. This is the Binomial Theorem, and it means that if you ever need to expand (a + b)¹⁰⁰, you don’t have to multiply anything out. You just read off Row 100 of the triangle.

Why does this work? Because expanding (a + b)ⁿ is secretly a counting problem. Each term in the expansion comes from choosing, for each of the n factors, whether to contribute an a or a b. The coefficient of aᵏbⁿ⁻ᵏ counts how many ways you can choose k factors to contribute an a — which is exactly C(n, k). Probability and algebra are speaking the same language.

Diagonals, Triangular Numbers, and Powers of Two

The deeper you look into Pascal’s Triangle, the more patterns emerge. Let me point out just a few.

The diagonals tell stories. The outermost diagonal is all 1s. The next diagonal in is 1, 2, 3, 4, 5, … — the counting numbers. The diagonal after that is 1, 3, 6, 10, 15, … — the triangular numbers, which count how many dots you need to make an equilateral triangle of side length n. (Think of bowling pins: 10 pins arranged in a triangle of side 4.) The next diagonal gives the tetrahedral numbers, which count stacked cannonballs.

The rows sum to powers of two. Add up all the numbers in any row: Row 0 gives 1, Row 1 gives 2, Row 2 gives 4, Row 3 gives 8, Row 4 gives 16. Every row doubles the previous one. This makes perfect sense: each row counts all possible outcomes of n coin flips, and every time you add a flip, you double the number of outcomes.

Colour the odd numbers. If you shade every odd number in Pascal’s Triangle and leave even numbers blank, a fractal emerges — specifically, the Sierpiński Triangle, one of the most famous fractals in mathematics. A simple rule about addition gives birth to infinite self-similar geometry. This is, to put it plainly, extraordinary.

The Real-World Payoff: Options Pricing

Pascal’s Triangle isn’t just a classroom curiosity. It sits at the heart of modern financial mathematics.

When a trader prices a stock option — the right to buy or sell a stock at a fixed price on or before a future date — they need to model all the possible paths the stock price could take. One classic model, the Binomial Options Pricing Model, imagines that at each time step, the stock price either goes up or goes down by a fixed factor. Sound familiar?

After n time steps, the number of paths that lead to exactly k “up” moves is C(n, k) — a number straight from Pascal’s Triangle. To find the fair price of the option, the trader weights each possible final payoff by its risk-neutral probability, with the path counts appearing inside those probabilities, and discounts back to today. The whole calculation is Pascal’s Triangle in disguise.

The famous Black-Scholes formula — part of the work for which Scholes and Merton later received the Nobel Memorial Prize in Economic Sciences — is essentially the continuous limit of this binomial model, taken as the time steps get smaller and smaller. The binomial distribution encoded by Pascal’s Triangle approaches the bell curve of probability. The humble triangle and the most important equation in finance are the same idea at different scales.

Your Mental Model to Keep

Here is the one insight I want you to carry away: counting and probability are the same thing.

Whenever you want to know the probability of some event happening k times out of n tries, you’re really asking two questions: how many ways can that happen, and how many total outcomes are there? Pascal’s Triangle answers the first question instantly. The second is almost always a power of 2 (for coin-like situations) or some other simple formula.

The next time you hear a statistic like “there’s a 1-in-10 chance of rain every day this week, so it’ll definitely rain at least once,” you can check it. The probability of no rain all week is (9/10)⁷ ≈ 0.478 — about 48%. So there’s actually a 52% chance of at least one rainy day. Not “definite.” Not even close to certain. Just slightly more likely than not.

Pascal’s Triangle won’t give you that answer directly — but the thinking behind it will. Count the outcomes. Weight them by their probabilities. Don’t let your intuition bully you into a wrong answer.

That’s the gift of the triangle: not just a set of numbers, but a way of seeing.