A bag has 2 red and 3 blue marbles, two are picked with replacement… Step by Step

Step by step

The question

Question: A bag has 2 red and 3 blue marbles.

Two marbles are picked, one after the other.

The first one is put back.

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

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: 5 marbles.

2 Draw a tree

We want red, then blue.

That is this path, in red.

3 Multiply along the branches

And means both happen.

We multiply along the path: 2/5, times 3/5.

P(red then blue) = 25 × 35

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(red then blue) = 2 × 35 × 5

2 times 3 is 6.

5 times 5 is 25.

So it is 6/25.

P(red then blue) = 625

4 Check

Let's check the tree.

Each end: multiply along its branches.

We do not cancel, so every bottom is 25.

All the ends together: 4 plus 6 plus 6 plus 9 is 25.

25/25 is 1.

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

425 + 625 + 625 + 925 = 2525 = 1

Let's look back.

First: independent or not.

Then: draw a tree.

Then: multiply along the branches.

Last: check.

Answer