Step by step
The question
Question: A bag has 2 green and 8 yellow marbles.
Two marbles are picked, one after the other.
The first one is not put back.
What is the probability both are yellow?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Independent or not
The first marble is not put back.
We call them dependent.
So the second pick has one marble fewer: 9.
And the colour picked first has one fewer.
2 Draw a tree
First pick: green is 2 out of 10, yellow is 8 out of 10.
After a green: 9 left, and 1 green and 8 yellow.
After a yellow: 9 left, and 2 green and 7 yellow.
We want yellow, then yellow.
That is this path, in red.
3 Multiply along the branches
And means both happen.
We get yellow 8 out of 10 times.
Of those times, 7 out of 9 go on to yellow.
A part of a part: of means times.
So we multiply along the path: 8/10, times 7/9.
Careful!
The first marble is not back.
The second pick is out of 9.
To multiply fractions, we do top times top, and bottom times bottom.
8 times 7 is 56.
10 times 9 is 90.
So it is 56/90.
4 Simplify
Both 56 and 90 can be divided by 2.
That makes it simpler.
So it is 28/45.
5 Check
Let's check the tree.
Each end: multiply along its branches.
We do not cancel, so every bottom is 90.
All the ends together: 2 plus 16 plus 16 plus 56 is 90.
90/90 is 1.
So the tree is right, and so is our path!
6 Pick the choice
Our answer is P(both yellow) = 28/45.
That is choice B.
7 Why not the others
A is 1/45.
Wrong: that is both green: 2/10 × 1/9.
C is 16/25.
Wrong: 8/10 × 8/10: after one yellow is taken, 7 of 9 are left.
That is the Careful mistake from before.
D is 4/5.
Wrong: only the first pick: 8 out of 10.
Let's look back.
First: independent or not.
Then: draw a tree.
Then: multiply along the branches.
Then: simplify.
Then: check.
Then: pick the choice.
Last: why not the others.