Question

In a family with 6 children, excluding multiple births, what is the probability of having exactly 1 girl?
Assume that having a boy is as likely as having a girl at each birth.

Answer 1

Start with what is the probability of having a girl out of 1 child?

Answer 2

1/1

Answer 3

That would be if a girl was a sure thing. Isn't it possible to have a boy or a girl for each birth?

Answer 4

right, 1/2

Answer 5

Good. Now how would you do if you wanted to the chances of having 2 girls?

Answer 6

would it just be 1/2 again because its 2/4? (Sorry, I've never done probabilities before now)

Answer 7

Close. No problem. That's why we are working up to the actual question.

All the possibilities for having 2 kids are: BB, GB, BG, GG.

So how likely is it to have both girls?

Answer 8

1/4

Answer 9

Exactly. And you can also do that by saying that the probability of having a girl the first child is 1/2 and the probability of having a girl the second child is 1/2 so the probability of both happening is 1/2 * 1/2 = 1/4.

Answer 10

We can flip this question and ask what is the probability of not having 2 girls too?

Answer 11

3/4

Answer 12

Good. This comes from 1 - 1/4. These numbers are not related by accident. If you know how likely something is, then the opposite (or how likely it is to not happen) is 1 - probability of it happening.

Answer 13

Okay, I understand that

Answer 14

Great. So getting back to the question. You need to figure out how many ways you can have exactly 1 girl out of 6 kids AND how many options for 6 kids you have all together. There are a bunch of ways to go about it, but you can start with figuring it out for 3 kids to get the method in an easier case.

Answer 15

Okay, so for three kids your choices are
BBB
BBG
BGG
GGG
GGB
GBG
BGB
GBB
so the probability of having one girl is 2/8, so 1/4?

Answer 16

Looks like there should be 3 to me. either the first child, second child or 3rd child.

Answer 17

right, so 3/8!

Answer 18

You have it there, but I think you missed it when you counted. No problem though, you are on the right path.

Let's see if you can take these results and figure out how many ways you could have exactly one girl with 6 kids?

Answer 19

Yes!

Answer 20

BBBBBB
BBBBBG
BBBBGB
BBBGBB
BBGBBB
BGBBBB
GBBBBB
BBBBGG
BBBGGG
BBGGGG
BGGGGG
GGGGGG
wait, is there a faster way to do this rather than write out every single option?

Answer 21

yes there is. In the case of 3 kids, there were 3 options because either the first kid was a girl, or the second, or the third and all the rest have to be boys. So for 6 kids ...

Answer 22

there's six chances, but how do i figure out the denominator? (I know it's 6/x but how do I know what x is?)

Answer 23

There's a pattern for that too. (I forgot it, but noticed it through the examples we did).

Answer 24

With 1 kid, there were 2 ways to have that child. With 2 kids there was 4 ways and with 3 kids there was 8.

Answer 25

To put it in a bit of a table:
# kids || # of possible outcomes for gender
1 || 2
2 || 4
3 || 8

How many possible outcomes for 4 kids?

Answer 26

16! Okay I think I've got it now, thank you!