Step by step
The question
Question: A bag has 4 red and 5 blue marbles.
2 marbles are picked at random.
What is the probability that 1 is a red and 1 is a blue?
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
9 C 2: the top 2 numbers from 9 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: 9 times 4.
That is 36 ways.
3 Count the ways we want
Picking 1 of 4 red: 4 ways.
Picking 1 of 5 blue: 5 ways.
Each choice of red goes with each choice of blue.
So we multiply: 4 times 5 is 20.
Careful!
We need both at the same time: red and blue.
And means multiply, not add.
4 Write the fraction
20 ways we want, out of 36.
5 Simplify
Both 20 and 36 can be divided by 4.
That makes it simpler.
So it is 5/9.
6 Check
Let's check another way: pick one at a time, in the order R, B.
Multiplied, that is 5/18.
But the 1 red and the 1 blue can come in 2 orders, each just as likely.
2 times that is 5/9.
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.