Step by step
The question
Question: A bag has 5 red and 3 blue marbles.
3 marbles are picked at random.
What is the probability that 2 are 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
8 C 3: the top 3 numbers from 8 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
6 on top and 3 times 2 at the bottom: 6 divided by 6 is 1.
What is left: 8 times 7.
That is 56 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.
Picking 1 of 3 blue: 3 ways.
Each choice of red goes with each choice of blue.
So we multiply: 10 times 3 is 30.
Careful!
We need both at the same time: red and blue.
And means multiply, not add.
4 Write the fraction
30 ways we want, out of 56.
5 Simplify
Both 30 and 56 can be divided by 2.
That makes it simpler.
So it is 15/28.
6 Check
Let's check another way: pick one at a time, in the order R, R, B.
Multiplied, that is 5/28.
But the 2 red and the 1 blue can come in 3 orders, each just as likely.
3 times that is 15/28.
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.
Then: simplify.
Last: check.