A bag has 4 red and 5 blue marbles, two are picked with replacement… Step by Step

Step by step

The question

Question: A bag has 4 red and 5 blue marbles.

Two marbles are picked, one after the other.

The first one is put back.

What is the probability the first is blue and the second is red?

1 Independent or not

The first marble is put back.

1st → does not change the 2nd

We call them independent.

independent:

the 1st does not change the 2nd

So the bag is the same for the second pick: 9 marbles.

2 Draw a tree

We want blue, then red.

That is this path, in red.

3 Multiply along the branches

And means both happen.

We multiply along the path: 5/9, times 4/9.

P(blue then red) = 59 × 49

and: multiply along the branches

Careful!

The marble is put back, so the second pick is just like the first.

Careful: put back: the same fractions again

To multiply fractions, we do top times top, and bottom times bottom.

P(blue then red) = 5 × 49 × 9

5 times 4 is 20.

9 times 9 is 81.

So it is 20/81.

P(blue then red) = 2081

4 Check

Let's check the tree.

Each end: multiply along its branches.

We do not cancel, so every bottom is 81.

All the ends together: 16 plus 20 plus 20 plus 25 is 81.

81/81 is 1.

So the tree is right, and so is our path!

1681 + 2081 + 2081 + 2581 = 8181 = 1

Let's look back.

First: independent or not.

Then: draw a tree.

Then: multiply along the branches.

Last: check.

Answer