Question

There are 11 pens in a box. 8 are black and 3 are red. Two pens are taken out without replacement. Work out the probability that two pens are the same colour? (4 marks)

Answer 1

0.563 chance

Explanation:

You need to make a probability tree diagram so you can work out the odds:

Overall you will end up with `8/11` (original amount of black pens) multiplied by `7/10` (amount of black pens left in the box) + `3/11` (overall amount of red pens) multiplied by `2/10` (amount of red pens left in the box).

This = 0.563 chance that you will pick 2 pens of the same colour, whether they be 2 black or 2 red.

Answer 2

`31/55`

Explanation:

There are two possibilities that we're trying to find: the probability to get `2` black pens and the probability to get `2` red pens. I'll start with the chance that you'll get two black pens.

The chance that the first pen you pick out of the box is black is `8/11`. The chance that the second pen you pick out of the box is `7/11` since you don't replace the first pen you took out of the box.

To figure out the chance that both the first and second pens you took out of the box were black, we multiply those two values together:

`8/11*7/10=56/110`

This is possibility number one. The second possibility that we want is the chance that you'll draw two red pens. To do this, we repeat the same process.

`3/11*2/10=6/110`

Now we know the chance you'll draw two black pens and the chance you'll draw two red pens at random. Since both of these are favourable outcomes, we add these two numbers together.

`56/110+6/110=62/110`

And finally, to simplify.

`62/110-:2=31/55`

As `31` is a prime number, we can't simplify further. Thus the answer is `31/55`. Or `0.563` (to 3 s.f.) as a decimal or `56%` (to 2 s.f.) as a percent.