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# A couple has two children, one of whom is a girl.

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A couple has two children, one of whom is a girl. If the probability [#permalink]

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05 Sep 2014, 03:58
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A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?

A. 1/8
B. 1/4
C. 1/3
D. 1/2
E. 2/3
[Reveal] Spoiler: OA
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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05 Sep 2014, 04:41
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AmoyV wrote:
A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?

A. 1/8
B. 1/4
C. 1/3
D. 1/2
E. 2/3

If the probability of having a boy or a girl is $$50%$$, the possibilities can be $$(BB, BG, GB, GG)$$

Now, it's given that one of the kids is a girl. So, the sample set reduces to $$(BG, GB, GG)$$

Probability of having two girls = $$\frac{1}{3}$$
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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06 Sep 2014, 08:03
Hi...
I don´t understand why permutation matters here. I think this problem is about combinatorics.
In the family there is already one girl. Then the probability of a second girl is 1/2.
Why does it matter whether the girl is the first or the second in the family?. I know that I am wrong but I don´t understand why.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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06 Sep 2014, 10:41
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Maxirosario2012 wrote:
Hi...
I don´t understand why permutation matters here. I think this problem is about combinatorics.
In the family there is already one girl. Then the probability of a second girl is 1/2.
Why does it matter whether the girl is the first or the second in the family?. I know that I am wrong but I don´t understand why.

The question never says that the first child is a girl. If it was so, then the probability would indeed be $$\frac{1}{2}$$.

But here we need to consider following two cases:

1) The first child is a girl
In this case there are two possibilities - $$GB, GG$$.

1) The first child is a boy
In this case there is just one possibility - $$BG$$.
Notice that $$BB$$ is not possible as the question explicitly states that one girl is a child.

Hence the probability is $$\frac{1}{3}$$.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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03 May 2015, 13:30
I still do not understand this... How is GB different than BG?

If you walk up to a couple and already know that A) they have two children and B) one is a girl....

then half the time they will have two girls and half the time they will have a boy and a girl...

Can someone confirm this? Seems like people are regurgitating information straight from the solutions instead of actually thinking through the problem.
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Posts: 44599
Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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04 May 2015, 02:40
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eltonj wrote:
A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?

A. 1/8
B. 1/4
C. 1/3
D. 1/2
E. 2/3

I still do not understand this... How is GB different than BG?

If you walk up to a couple and already know that A) they have two children and B) one is a girl....

then half the time they will have two girls and half the time they will have a boy and a girl...

Can someone confirm this? Seems like people are regurgitating information straight from the solutions instead of actually thinking through the problem.

First child ---- Second child:
B ---------------------------- B;
B ---------------------------- G;
G ---------------------------- B;
G ---------------------------- G.

We know that one if the kids is a girl. So, we have one of the three cases in red. Each case there is equally likely (each has the probability of 1/2*1/2=1/4). So, the probability of GG is 1/3.

Hope it's clear.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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04 May 2015, 09:06
Bunuel wrote:
eltonj wrote:
A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?

A. 1/8
B. 1/4
C. 1/3
D. 1/2
E. 2/3

I still do not understand this... How is GB different than BG?

If you walk up to a couple and already know that A) they have two children and B) one is a girl....

then half the time they will have two girls and half the time they will have a boy and a girl...

Can someone confirm this? Seems like people are regurgitating information straight from the solutions instead of actually thinking through the problem.

First child ---- Second child:
B ---------------------------- B;
B ---------------------------- G;
G ---------------------------- B;
G ---------------------------- G.

We know that one if the kids is a girl. So, we have one of the three cases in red. Each case there is equally likely (each has the probability of 1/2*1/2=1/4). So, the probability of GG is 1/3.

Hope it's clear.

Hello, thank you kindly for your response. I fully understand how an answer of 1/3 is found. That being said, I am still not sure why BG and GB is being treated as two separate cases. In my mind, the problem looks like this:

Child Child
B ----------- B
B ----------- G
G ----------- G

I don't see the need to have BG and GB. This problem has nothing to do with the order of the children. We know they have a Girl, so they either have a boy and a girl or two girls.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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05 May 2015, 00:58
eltonj wrote:
Bunuel wrote:
eltonj wrote:
A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?

A. 1/8
B. 1/4
C. 1/3
D. 1/2
E. 2/3

I still do not understand this... How is GB different than BG?

If you walk up to a couple and already know that A) they have two children and B) one is a girl....

then half the time they will have two girls and half the time they will have a boy and a girl...

Can someone confirm this? Seems like people are regurgitating information straight from the solutions instead of actually thinking through the problem.

First child ---- Second child:
B ---------------------------- B;
B ---------------------------- G;
G ---------------------------- B;
G ---------------------------- G.

We know that one if the kids is a girl. So, we have one of the three cases in red. Each case there is equally likely (each has the probability of 1/2*1/2=1/4). So, the probability of GG is 1/3.

Hope it's clear.

Hello, thank you kindly for your response. I fully understand how an answer of 1/3 is found. That being said, I am still not sure why BG and GB is being treated as two separate cases. In my mind, the problem looks like this:

Child Child
B ----------- B
B ----------- G
G ----------- G

I don't see the need to have BG and GB. This problem has nothing to do with the order of the children. We know they have a Girl, so they either have a boy and a girl or two girls.

Having an older girl and a younger boy is different from having an older boy and a younger girl.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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05 May 2015, 09:47
Age is 100% irrelevant in this question, all we care about is gender. The answer is 1/2.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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05 May 2015, 09:55
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Expert's post
eltonj wrote:
Age is 100% irrelevant in this question, all we care about is gender. The answer is 1/2.

The correct answer is 1/3, not 1/2. Having a boy and a girl is twice as likely than having two boys (or two girls), 1/2 and 1/4 respectively, so you should take this into account.
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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05 May 2015, 10:14
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Bunuel wrote:
eltonj wrote:
Age is 100% irrelevant in this question, all we care about is gender. The answer is 1/2.

The correct answer is 1/3, not 1/2. Having a boy and a girl is twice as likely than having two boys (or two girls), 1/2 and 1/4 respectively, so you should take this into account.

Ahhh okay this makes much more sense to me. Thank you very much!
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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14 Jul 2015, 06:12
Bunuel wrote:
eltonj wrote:
Age is 100% irrelevant in this question, all we care about is gender. The answer is 1/2.

The correct answer is 1/3, not 1/2. Having a boy and a girl is twice as likely than having two boys (or two girls), 1/2 and 1/4 respectively, so you should take this into account.

I was having the same problem as eltonj - but after reading through this thread, Bunuel's last explanation cleared it up. Thank's Bunuel.

Here's an interesting link that shows an interesting visual explaining conditional probability: http://setosa.io/conditional/
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Re: A couple has two children, one of whom is a girl. If the probability [#permalink]

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14 Jul 2015, 06:47
Sample space = {bb,bg,gb,gg}

Favourable event = {gg}.

Since it's given that one of the child is a girl, the new Sample Space = {bg,gb,gg}.

So, probability = 1/3. Ans (C).
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Re: A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 04:56
REDWASP2205 wrote:
A couple has two children, one of whom is a girl. If the probability of having a girl or a boy is 50%, what is the probability that the couple has two daughters?
A) 1/8
B) 1/4
C) 1/3
D) 1/2
E) 2/3

Hi,
we know one is girl..

the cases possible with one kid as girl is
BG or GB or GG -- 3 ways
we are talking of prob of GG -- 1 way
ans $$\frac{1}{3}$$
C
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Re: A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 05:26
chetan2u

But knowing One of them is a girl, shouldn't we remove BG from the possible cases.
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A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 05:28
REDWASP2205 wrote:
chetan2u

But knowing One of them is a girl, shouldn't we remove BG from the possible cases.

No, Because we just know that one of them is G..
here the order is important but we are not given the order but just that one is a G..
If it said the first one is a G, then yes we would have had ONLY two cases GB or GG..

Merged topics
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Re: A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 06:13
chetan2u wrote:
REDWASP2205 wrote:
chetan2u

But knowing One of them is a girl, shouldn't we remove BG from the possible cases.

No, Because we just know that one of them is G..
here the order is important but we are not given the order but just that one is a G..
If it said the first one is a G, then yes we would have had ONLY two cases GB or GG..

Hi chetan2u,

This one is crucial for me to understand. whether a boy is first and girl is second or boy is younger than girl or vice versa--WHY do we consider BG, GB as two cases instead of ONE case only. We are not arranging here.

I looked at the responses from Bunuel and others on original post too but i still have this doubt. Could you please explain this?
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Re: A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 06:24
HKD1710 wrote:
chetan2u wrote:
REDWASP2205 wrote:
chetan2u

But knowing One of them is a girl, shouldn't we remove BG from the possible cases.

No, Because we just know that one of them is G..
here the order is important but we are not given the order but just that one is a G..
If it said the first one is a G, then yes we would have had ONLY two cases GB or GG..

Hi chetan2u,

This one is crucial for me to understand. whether a boy is first and girl is second or boy is younger than girl or vice versa--WHY do we consider BG, GB as two cases instead of ONE case only. We are not arranging here.

I looked at the responses from Bunuel and others on original post too but i still have this doubt. Could you please explain this?

Hi,

I would relate this Q to two different Q.

1) Ways to pick two books out of 4 books randomly?
4C2..

2) Ways to pick two books out of 4 books one after another?
this will be 4C2 *2!

Our Q is similar to the 2nd case
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A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 10:45
HKD1710 wrote:
chetan2u wrote:
REDWASP2205 wrote:
chetan2u

But knowing One of them is a girl, shouldn't we remove BG from the possible cases.

No, Because we just know that one of them is G..
here the order is important but we are not given the order but just that one is a G..
If it said the first one is a G, then yes we would have had ONLY two cases GB or GG..

Hi chetan2u,

This one is crucial for me to understand. whether a boy is first and girl is second or boy is younger than girl or vice versa--WHY do we consider BG, GB as two cases instead of ONE case only. We are not arranging here.

I looked at the responses from Bunuel and others on original post too but i still have this doubt. Could you please explain this?

Hello HKD1710

Please go Through eshan429 's post

probability-217147.html#p1406381

IMHO, I don't feel you need to consider younger /older in this case. The chances of having a twins ( Boy and Girl) in a single delivery can not be rejected.

Both Boy and Girl may be born at the same time, we need to consider only the probability of having only 2 daughters ( They may be twins as well - Born on the same day)

Younger / Elder daughter is not important in this case , we simply restrict ourself to a female child (daughter)

Happy Preparations

Abhishek
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Re: A couple has two children, one of whom is a girl. [#permalink]

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23 Apr 2016, 10:53
Thanks Abhishek009,

I went thru all those solutions and i still did not understand only the following:

Quote:
WHY do we consider BG, GB as two cases instead of ONE case only. We are not arranging here.

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Re: A couple has two children, one of whom is a girl.   [#permalink] 23 Apr 2016, 10:53

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