Page 140 - Probability Demystified

CHAPTER 7 The Binomial Distribution 129

1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1
Each number in the triangle is the sum of the number above and to the
right of it and the number above and to the left of it. For example, the
number 10 in the fifth row is found by adding the 4 and 6 in the fourth row.
The number 15 in the sixth row is found by adding the 5 and 10 in the
previous row.
The numbers in each row represent the number of different outcomes when
coins are tossed. For example, the numbers in row 3 are 1, 3, 3, and 1. When
3 coins are tossed, the outcomes are

THH HTT
HTH THT

HHH HHT TTH TTT
Notice that there is only one way to get 3 heads. There are 3 different ways
to get 2 heads and a tail. There are 3 different ways to get two tails and a head
and there is one way to get 3 tails. The same results apply to the genders of
the children in a family with three children.
Another property of the triangle is that it represents the answer to the
number of combination of n items taking r items at a time as shown.
C
0 0
C C
1 0 1 1
C C C
2 0 2 1 2 2

C C C C
3 0 3 1 3 2 3 3
etc: