A bag has 3 red and 5 blue marbles, two are picked without… 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 not put back.

What is the probability both are red?

1 Independent or not

The first marble is not put back.

1st → changes the 2nd

We call them dependent.

dependent:

the 1st changes the 2nd

So the second pick has one marble fewer: 7.

And the colour picked first has one fewer.

2nd pick = 8 − 1 = 7 marbles left

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 2/7.

P(both red) = 38 × 27

and: multiply along the branches

Careful!

The first marble is not back.

The second pick is out of 7.

Careful: not 38 × 38 ✗

Before we multiply, we make the numbers smaller.

2 on top and 8 at the bottom both divide by 2.

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

P(both red) = 3 × 14 × 7

3 times 1 is 3.

4 times 7 is 28.

So it is 3/28.

P(both red) = 328

4 Check

Let's check the tree.

Each end: multiply along its branches.

We do not cancel, so every bottom is 56.

All the ends together: 6 plus 15 plus 15 plus 20 is 56.

56/56 is 1.

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

656 + 1556 + 1556 + 2056 = 5656 = 1

Let's look back.

First: independent or not.

Then: draw a tree.

Then: multiply along the branches.

Last: check.

Answer