A bag has 3 red and 5 blue marbles, two are picked with replacement… Step by Step

Step by step

The question

Question: A bag has 3 red and 5 blue marbles.

Two marbles are picked, one after the other.

The first one is put back.

What is the probability both are 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: 8 marbles.

2 Draw a tree

We want red, then red.

That is this path, in red.

3 Multiply along the branches

And means both happen.

We multiply along the path: 3/8, times 3/8.

P(both red) = 38 × 38

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(both red) = 3 × 38 × 8

3 times 3 is 9.

8 times 8 is 64.

So it is 9/64.

P(both red) = 964

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: 9 plus 15 plus 15 plus 25 is 64.

64/64 is 1.

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

964 + 1564 + 1564 + 2564 = 6464 = 1

Let's look back.

First: independent or not.

Then: draw a tree.

Then: multiply along the branches.

Last: check.

Answer