Step by step
The question
Question: A die is rolled twice.
What is the probability of getting a number more than 4, then a number less than 3?
1 Independent or not
The first roll does not change the second.
We call them independent.
It is the same die, with the same 6 sides.
2 Draw a tree
We want a number more than 4, then a number less than 3.
That is this path, in red.
3 Multiply along the branches
And means both happen.
We multiply along the path: 2/6, times 2/6.
Careful!
And means multiply, not add.
Before we multiply, we make the numbers smaller.
2 on top and 6 at the bottom both divide by 2.
2 on top and 6 at the bottom both divide by 2.
To multiply fractions, we do top times top, and bottom times bottom.
1 times 1 is 1.
3 times 3 is 9.
So it is 1/9.
4 Check
Let's check the tree.
Each end: multiply along its branches.
We do not cancel, so every bottom is 36.
All the ends together: 4 plus 8 plus 8 plus 16 is 36.
36/36 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.