Step by step
The question
Question: A club has 5 boys and 3 girls.
2 of them are picked at random.
What is the probability that both are girls?
The probability is the number of ways we want, over all the ways.
P stands for probability.
1 Order: yes or no
Does order matter?
No.
Picking Ann, then Bo, is the same group as Bo, then Ann.
So we count groups: combinations.
2 Count all the ways
8 C 2: the top 2 numbers from 8 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
8 on top and 2 at the bottom: 8 divided by 2 is 4.
What is left: 4 times 7.
That is 28 ways.
3 Count the ways we want
3 C 2: the top 2 numbers from 3 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
2 on top and 2 at the bottom: 2 divided by 2 is 1.
So there are 3 ways.
4 Write the fraction
3 ways we want, out of 28.
Careful!
The top and the bottom must count the same way: both are groups, with C.
5 Check
Let's check another way: pick one at a time.
The first: 3 out of 8.
Then one fewer of each.
Multiplied, that is 3/28.
The same.
So the answer is right!
Let's look back.
First: order: yes or no.
Then: count all the ways.
Then: count the ways we want.
Then: write the fraction.
Last: check.