Question

15 names are in a hat, 6 girls and 9 boys. What is the probability of drawing, without replacement, two girls names in a row?

Answer 1

`1/7`

Explanation:

There is a total of 15 names in the hat. `6/15` are girls and `9/15` are boys.

First, you pull one name. The odds of this name being a girl are `6/15`.

Next, you'll pull out a second name but the odds have changed.

The new odds of drawing a girls' name are `5/14` and the new odds of drawing a boys' name are `9/14`. You do this because there are not as many names in the hat as there were before, hence the subtraction of 1 from the denominator. There is also one fewer girls' names in the hat because you pulled one out.

The odds of you drawing a second girls' name is `5/14`.

Take both of these odds (`6/15` and `5/14`) and multiply them together. This will give you the odds of drawing two girls' names in a row without replacement.

The final answer should be `30/210` which simplifies to `1/7`.

Answer 2

`C_(6,2)/C_(15,2)=1/7`

Explanation:

An alternate way to work this is to calculate the number of ways 2 girls's names can be pulled and divide that by the total number of ways 2 names can be pulled. We're working with combinations (we don't care in what order the names are pulled) and the general formula is:

`C_(n,k)=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

So we'll have:

`C_(6,2)/C_(15,2)=((6!)/(2!4!))/((15!)/(2!13!))=((6!)/(2!4!))((2!13!)/(15!))=>`

`=>(6xx5xx4!)/(4!)(13!)/(15xx14xx13!)`

`=> 30/210=1/7`