Step by step
The question
Question: A bag has 5 red and 4 blue marbles.
3 marbles are picked at random.
What is the probability that all 3 are red?
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 3: the top 3 numbers from 9 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
9 and 3 both divide by 3: 3.
8 and 2 both divide by 2: 4.
What is left is 3 times 4 times 7.
3 times 4 is 12.
12 times 7 is 84.
So there are 84 ways.
3 Count the ways we want
5 C 3: the top 3 numbers from 5 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
3 and 3 both divide by 3: 1.
4 and 2 both divide by 2: 2.
What is left: 5 times 2.
That is 10 ways.
4 Write the fraction
10 ways we want, out of 84.
Careful!
The top and the bottom must count the same way: both are groups, with C.
5 Simplify
Both 10 and 84 can be divided by 2.
That makes it simpler.
So it is 5/42.
6 Check
Let's check another way: pick one at a time.
The first: 5 out of 9.
Then one fewer of each.
Multiplied, that is 5/42.
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.