Step by step
The question
Question: A club has 5 boys and 5 girls.
4 of them are picked at random.
What is the probability that 2 are boys and 2 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
10 C 4: the top 4 numbers from 10 down, over 4 factorial.
Before we multiply, we make the numbers smaller.
8 and 4 both divide by 4: 2.
9 and 3 both divide by 3: 3.
10 and 2 both divide by 2: 5.
What is left is 5 times 3 times 2 times 7.
5 times 3 is 15.
15 times 2 is 30.
30 times 7 is 210.
So there are 210 ways.
3 Count the ways we want
5 C 2: the top 2 numbers from 5 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
4 on top and 2 at the bottom: 4 divided by 2 is 2.
What is left: 5 times 2.
That is 10 ways.
5 C 2: the top 2 numbers from 5 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
4 on top and 2 at the bottom: 4 divided by 2 is 2.
What is left: 5 times 2.
That is 10 ways.
Each choice of boys goes with each choice of girls.
So we multiply: 10 times 10 is 100.
Careful!
We need both at the same time: boys and girls.
And means multiply, not add.
4 Write the fraction
100 ways we want, out of 210.
5 Simplify
Both 100 and 210 can be divided by 10.
That makes it simpler.
So it is 10/21.
6 Check
Let's check another way: pick one at a time, in the order B, B, G, G.
Multiplied, that is 5/63.
But the 2 boys and the 2 girls can come in 6 orders, each just as likely.
6 times that is 10/21.
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.
Then: simplify.
Last: check.