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 both 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 one marble, then another, gives the same marbles as the other way round.
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
4 C 2: the top 2 numbers from 4 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: 2 times 3.
That is 6 ways.
4 Write the fraction
6 ways we want, out of 36.
Careful!
The top and the bottom must count the same way: both are groups, with C.
5 Simplify
Both 6 and 36 can be divided by 6.
That makes it simpler.
So it is 1/6.
6 Check
Let's check another way: pick one at a time.
The first: 4 out of 9.
Then 3 out of 8: one fewer of each.
Multiplied, that is 1/6.
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.