A bag has 5 white and 5 black marbles, two are picked without… Step by Step

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The question

Question: A bag has 5 white and 5 black marbles.

Two marbles are picked, one after the other.

The first one is not put back.

What is the probability both are black?

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: 9.

And the colour picked first has one fewer.

2nd pick = 10 − 1 = 9 marbles left

2 Draw a tree

We want black, then black.

That is this path, in red.

3 Multiply along the branches

And means both happen.

We multiply along the path: 5/10, times 4/9.

P(both black) = 510 × 49

and: multiply along the branches

Careful!

The first marble is not back.

The second pick is out of 9.

Careful: not 510 × 510 ✗

Before we multiply, we make the numbers smaller.

4 on top and 10 at the bottom both divide by 2.

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

P(both black) = 5 × 25 × 9

5 times 2 is 10.

5 times 9 is 45.

So it is 10/45.

P(both black) = 1045

4 Simplify

Both 10 and 45 can be divided by 5.

That makes it simpler.

P(both black) = 10 ÷ 545 ÷ 5

So it is 2/9.

P(both black) = 29

5 Check

Let's check the tree.

Each end: multiply along its branches.

We do not cancel, so every bottom is 90.

All the ends together: 20 plus 25 plus 25 plus 20 is 90.

90/90 is 1.

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

2090 + 2590 + 2590 + 2090 = 9090 = 1

Let's look back.

First: independent or not.

Then: draw a tree.

Then: multiply along the branches.

Then: simplify.

Last: check.

Answer