Quick Answer: Pascal's Triangle is a geometric arrangement of numbers where each number is the sum of the two directly above it. In probability, the rows of Pascal's Triangle tell you exactly how many combinations exist for coin flips, genetics, or any binary (two-outcome) event.
Building the Triangle
Start with 1 at the top. The next row is 1, 1. The next is 1, 2, 1 (because 1+1=2). The next is 1, 3, 3, 1 (because 1+2=3, 2+1=3).
- 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
Coin Flips and Pascal's Triangle
Each row gives you the exact number of outcomes when flipping coins. Look at Row 3 (1, 3, 3, 1). If you flip 3 coins, what are the possible outcomes? Add the row up: 1+3+3+1 = 8 total outcomes.
- 1 way to get exactly 0 Heads (TTT)
- 3 ways to get exactly 1 Head (HTT, THT, TTH)
- 3 ways to get exactly 2 Heads (HHT, HTH, THH)
- 1 way to get exactly 3 Heads (HHH)
Combinations (nCr)
The triangle also calculates combinations. "Choose 2 items from a group of 4" is written as 4C2. Go to Row 4. Count across starting from 0: the 0th item is 1, the 1st is 4, the 2nd is 6. So, 4C2 = 6. There are 6 ways to choose 2 items from 4.
The Binomial Distribution
If you want to know the probability of getting exactly 2 heads in 4 flips, find the number of ways it can happen (Row 4, item 2 = 6), and divide by the total possible outcomes for that row (1+4+6+4+1 = 16). The probability is 6/16, or 37.5%.