Step by step
The question
Question: A bag has 4 red and 4 blue marbles.
Two marbles are picked, one after the other.
The first one is put back.
What is the probability both are blue?
1 Independent or not
The first marble is put back.
We call them independent.
So the bag is the same for the second pick: 8 marbles.
2 Draw a tree
We want blue, then blue.
That is this path, in red.
3 Multiply along the branches
And means both happen.
We multiply along the path: 4/8, times 4/8.
Careful!
The marble is put back, so the second pick is just like the first.
Before we multiply, we make the numbers smaller.
4 on top and 8 at the bottom both divide by 4.
4 on top and 8 at the bottom both divide by 4.
To multiply fractions, we do top times top, and bottom times bottom.
1 times 1 is 1.
2 times 2 is 4.
So it is 1/4.
4 Check
Let's check the tree.
Each end: multiply along its branches.
We do not cancel, so every bottom is 64.
All the ends together: 16 plus 16 plus 16 plus 16 is 64.
64/64 is 1.
So the tree is right, and so is our path!
Let's look back.
First: independent or not.
Then: draw a tree.
Then: multiply along the branches.
Last: check.