Question

example 1 find all possible outcomes
hailey has two sisters and no brothers. josh has
two brothers and no sisters. they wonder what the
chances are, in a family with three children, that the
children will be all boys or all girls. how can they
determine all possible combinations of boys and
girls in a family with three children?

step 1 list the different events.
child 1 is either a boy or a girl.

child 2 is either a boy or a girl.

child 3 is either a boy or a girl.

this is a compound event. a compound event
consists of two or more events. this compound event
consists of three events.

step 2 make a tree diagram to represent the sample
space. a tree diagram shows all the possible outcomes.

bbb bbg bgb bgg gbb gbg ggb ggg
hailey and josh can make a tree diagram to show
the sample space of boys and girls in a family with
three children.

Explanation:

Step1: Define each child's outcome

Each child has 2 possible outcomes: Boy (B) or Girl (G)

Step2: Enumerate all combinations

Combine each child's outcome systematically:

1. Child1=B, Child2=B, Child3=B → BBB
2. Child1=B, Child2=B, Child3=G → BBG
3. Child1=B, Child2=G, Child3=B → BGB
4. Child1=B, Child2=G, Child3=G → BGG
5. Child1=G, Child2=B, Child3=B → GBB
6. Child1=G, Child2=B, Child3=G → GBG
7. Child1=G, Child2=G, Child3=B → GGB
8. Child1=G, Child2=G, Child3=G → GGG

Answer:

All possible combinations are: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG
The probability of all boys is $\frac{1}{8}$, and the probability of all girls is $\frac{1}{8}$.