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.