Step by step
The question
Question: A class has 7 boys and 5 girls.
2 of them are picked at random.
What is the probability that 1 is a boy and 1 is a girl?
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
12 C 2: the top 2 numbers from 12 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
12 on top and 2 at the bottom: 12 divided by 2 is 6.
What is left: 6 times 11.
That is 66 ways.
3 Count the ways we want
Picking 1 of 7 boys: 7 ways.
Picking 1 of 5 girls: 5 ways.
Each choice of boys goes with each choice of girls.
So we multiply: 7 times 5 is 35.
Careful!
We need both at the same time: boys and girls.
And means multiply, not add.
4 Write the fraction
35 ways we want, out of 66.
5 Check
Let's check another way: pick one at a time, in the order B, G.
Multiplied, that is 35/132.
But the 1 boy and the 1 girl can come in 2 orders, each just as likely.
2 times that is 35/66.
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.