## February 26, 2011

### Two Daughters, Three Questions

1. A couple has two children, what are the odds they are both girls?

2. Given that one of their two children is a girl, what are the odds
the other child is a girl?

3. Given that one is a girl named "Flo", what are the odds the other is
a girl?

Kris Lucius said...
This comment has been removed by the author.
Pierce said...

1. 50%
2. 25%
3. 25%

Anonymous said...

1. 50%
2. 50%
3. 50%

Dont all the questions ask the same thing ?

Rajesh said...

1.33.33%
2.50%
3.50%

Anonymous said...

I posted the question and no one is correct. I'll allow this to brew for a while... It is more difficult than it appears.

nune said...

1. 1/4
2. 1/3
3. 1/3

Gaurav Aggarwal said...

1. 25%
2. 50%
3. 50%

prat said...

Assumption.. No judwa/sixer

events .. Younger/Older

Girl/Girl
Girl/Boy
Boy/Girl
Boy/Boy

1. A couple has two children, what are the odds they are both girls?
clearly 1 out of 4 events ..

2. Given that one of their two children is a girl, what are the odds the other child is a girl?

we shud remove the Boy/Boy event here ..
so left with 3 events ..G/G out of three events ..

3. Given that one is a girl named "Flo", what are the odds the other is a girl?

similarly to Q2 ..we shud remove the Boy/Boy event here ..
so left with 3 events ..G/G out of three events ..

Dogtag114 said...

Prat is closest with the first two right. #3 needs some work. It is not the same answer as #2.

nune said...

I think 2 and 3 have same answers: 1/3, as the detail that one girl is named "Flo" gives us nothing.

Anonymous said...

25 percent
50 percent
50 percent

João Marques said...

beiing GF a Girl named Flo, and beeing GNF a Girl not named Flo, the set of possible events is
(GF, GF)
(GF, B)
(GF, GNF)
(GNF, GF)
(B,GF)

The odd of having two girls is 3/5 (60%)

Rajesh said...

1.25%
2.33.33%
3.50%

Secret Squïrrel said...

I agree, Prat is correct for the first two.

Q1 is clearly 1/4.

Q2 is 1/3. Prat has already explained this but here is another way to look at this.

Rather than starting with the statement "... that one of their two children is a girl, ..." and trying to work the odds out, start instead with working out what events would lead to that statement being true and then calculate their (the events') odds.

If they had 2 boys you could not truthfully say that one of them is a girl (1/4).
If they had a boy then a girl it would be true (1/4).
Same if they had a girl then a boy (1/4).
Lastly, it's also true if they have 2 girls (1/4).

Of the three equally probable cases where the statement could be truthfully made, in only one is the other child also a girl.

So the answer to Q2 is 1/3.

Question 3
However, labelling one of the children, either by name as here (Flo) or by any other identifiable character (younger, taller, green-fingered, happy, etc) changes everything.

You can't simply replace the word 'girl' with 'Flo' and then calculate the odds to be the same as for Q2. This is because 'girl' is a class or category name and 'Girl A ' is exactly the same as, and interchangeable with, 'Girl B'. However, 'Flo' is a specific member of the class 'Girl'" and cannot be interchanged with 'NotFlo'.

Here are the possible cases using the 'Flo' labelling system. It might be easier to understand if we replace 'Flo' with 'elder' and 'NotFlo' with 'younger' but it doesn't matter which labelling system we use.

FLO|NotFLO
.B.|.B.
.B.|.G.
.G.|.B.
.G.|.G.

The first two cases are eliminated from consideration since we are told that it is a girl named 'Flo', not a boy.

Of the only two cases to be considered, in only one is the other child also a girl.

So the answer to Q3 is 1/2.

Of course, all this assumes 3 things which aren't actually true in the real world.

Assumption 1: That the ratio of boys to girls is exactly 1:1.

It isn't and varies by geographic region.

Assumption 2: That the sex of one child does not influence the sex of the other.

If the two children are twins then, if you know the sex of one, you automatically know the sex of the other. If they are identical twins, the sexes are the same, if they're fraternal (dizygotic) twins, the sexes are different.

Assumption 3: That 'boy' and 'girl' are the only possible cases.

This ignores the fact that some people are true hermaphrodites (both male and female) or are of indeterminate sex (intersex).

Dogtag114 said...

Well Done. Some people will argue the answer of question #3 once they understand question #2.

Anonymous said...

Since the question doesn't specify older or younger, wouldn't there only be 3 options:BB,BG and GG?
Then the answers are:#1 is 1/3; #2 is 1/2; #3 is 1/2.

Anonymous said...

wrong!

3) Is 1/3. Fault in logic is that there is no reason to assume equally likely probabilities in this table:

FLO|NotFLO
.B.|.B.
.B.|.G.
.G.|.B.
.G.|.G.

in this one (we can reasonably assume) all outcomes are 25%

events .. Younger/Older

Girl/Girl
Girl/Boy
Boy/Girl
Boy/Boy

If it was Taller/Shorter

Girl/Girl 25%
Girl/Boy <25%
Boy/Girl >25%
Boy/Boy 25%

Since we know boys are generally taller (A couples children could be teens or full grown adults), If we were told the taller child is a girl, then it would be more likely the other child was also a girl.

A girl named Flo gives no information. So 1/3 remains correct.

This is a cool site but answers should really be checked somehow!

Anonymous said...

Why would you put BG and GB as two separate events? They are the same event. They both equate to a boy and a girl thus eliminating one of the events.

BB
GB
GG

Since one is a girl, we can now eliminate BB.
Now we are left with GB and GG
Odds of them having two girls when we already know that they have one girl is 50%

The answer is 1/2 or 50%

John G said...

The sample space for having two children is S = {GG, GB, BG, BB}

1. Probability they are both girls = 1/4

2. Probability that other is a girl given one is a girl = 1/3

3. Probability that other is a girl give that one is a girl named Flo = 1/4

Anonymous said...

1/4
1/2
1/2

Anonymous said...

1 Flo, girl
2 girl, Flo
3 Flo, boy
4 boy, Flo

so 1/2

Sophia said...

The sample space for having two children is S = {GG, GB, BG, BB} 1. Probability they are both girls = 1/4 2. Probability that other is a girl given one is a girl = 1/3 3. Probability that other is a girl give that one is a girl named Flo = 1/4

Dogtag114 said...

Ran into this today... I was the original poster.