A bag has 2 green and 5 yellow marbles, two are picked without… Step by Step

Step by step

The question

Question: A bag has 2 green and 5 yellow marbles.

Two marbles are picked, one after the other.

The first one is not put back.

What is the probability both are green?

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: 6.

And the colour picked first has one fewer.

2nd pick = 7 − 1 = 6 marbles left

2 Draw a tree

We want green, then green.

That is this path, in red.

3 Multiply along the branches

And means both happen.

We multiply along the path: 2/7, times 1/6.

P(both green) = 27 × 16

and: multiply along the branches

Careful!

The first marble is not back.

The second pick is out of 6.

Careful: not 27 × 27 ✗

Before we multiply, we make the numbers smaller.

2 on top and 6 at the bottom both divide by 2.

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

P(both green) = 1 × 17 × 3

1 times 1 is 1.

7 times 3 is 21.

So it is 1/21.

P(both green) = 121

4 Check

Let's check the tree.

Each end: multiply along its branches.

We do not cancel, so every bottom is 42.

All the ends together: 2 plus 10 plus 10 plus 20 is 42.

42/42 is 1.

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

242 + 1042 + 1042 + 2042 = 4242 = 1

Let's look back.

First: independent or not.

Then: draw a tree.

Then: multiply along the branches.

Last: check.

Answer