Probability Master · Lesson 3

Pascal's Amazing Triangle

One tiny rule — every number is the sum of the two above it — builds a triangle stuffed with hidden patterns. And its deepest secret? It's been counting your coin flips all along.

Unit 6 · Lesson 3 of 4 25 XP available

🔺 One rule builds everything

Start with a 1 at the top. Every other number is the sum of the two numbers above it (the edges only have one number above, so they stay 1). That's the whole recipe — a French mathematician named Blaise Pascal made this triangle famous in the 1650s, and mathematicians are still finding new patterns in it.

Activity 1: Build It Yourself 🧱

Rows 0–2 are done. You fill in rows 3, 4, and 5 — type each number (remember: sum of the two above). Correct answers lock in green.

15 cells to go…

Activity 2: Pattern Hunt 🔍

Now use your triangle (grow it first for the best view!). Three treasures are hiding:

1. Add up each row: 1, 2, 4, 8, 16… What's the pattern of the row sums?

They go up by 2 each time They DOUBLE — powers of 2! They're the counting numbers

2. Look at the second diagonal (just inside the 1s edge): 1, 2, 3, 4, 5…

The counting numbers, in order Random numbers Prime numbers

3. Fold the triangle down the middle. What happens?

Nothing special The two halves match — it's perfectly symmetric The left side is always bigger

Activity 3: The Coin Flip Secret 🪙

Here's the mind-bender. Flip 4 coins. Row 4 of your triangle — 1, 4, 6, 4, 1 — counts exactly how many ways to get each number of heads:

0 heads

1 head

2 heads

3 heads

4 heads

1. How many ways can 4 flips land with exactly 2 heads?

4 ways 6 ways — the middle of row 4 2 ways — HH first or HH last

2. How many total outcomes do 4 flips have? (Hint: add up the whole row… or remember the row-sum pattern!)

8 16 — the row sum, which is 2×2×2×2 24

3. So P(exactly 2 heads in 4 flips) = ?

6/16 — more likely than any other single count, but NOT most of the time 1/2 — heads is 50%, so half 6/10

🏆 Your XP Progress

○ Built rows 3–5 yourself +10 XP
○ Found all three patterns +8 XP
○ Unlocked the coin flip secret +7 XP

Total XP earned: 0 / 25

🖨️ Take the triangle home

Download the Triangle Kit — a blank triangle to fill by hand (rows 0–8!), a pattern-hunt checklist, and the coin-flip challenge on paper.

🧭 Keep Going

← Lesson 1: The Coin Flip Lab Where the flips began. 🏠 Probability Master Hub Your training program.

Lesson 2 — The Dice Detective — and Lesson 4 are coming soon!

🏆 Complete This Lesson!

Triangle mastered? Mark this lesson complete to track your progress.

✅ Lesson 3 complete — you built a 370-year-old mathematical treasure and stole its secret. Pascal would be delighted.