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# One Girl - One Boy

Best Answer Riddari, 13 July 2007 - 07:38 PM

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348 replies to this topic

### #21 skale

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Posted 17 July 2007 - 04:04 PM

If these are the possible combinations and you eliminated the boy-boy combo that means that you are solving this combo problem backwards. What I am trying to say is that since the untouched chances are 50% 50% because:
1st child / 2nd child
Girl / Girl
Girl / Boy
Boy / Girl
Boy / Boy
so for 2nd child the set of possibilities is {girl, boy, girl, boy} which gives 2 out of 4 for girls and gives you 50%.
YOU HAVE TO eliminate the 1st child boy, 2nd child girl because the 1st child is already known to be a girl.

THEREFORE: your only possible outcomes are girl girl and girl boy which gives you 50% 50%.

No, the question says:
They have two kids, one of them is a girl, what is the probability that the other kid is also a girl.
Let me explain it with the coin example, there are 4 combinations flipping 2 coins:
1) HH
2) HT
3) TH
4) TT

Now what is the probability of one HEAD and one TAIL, clearly it is 2/4. Now let us say that you know one of the toss is a head (and not that the first toss is head). What is the possibility of one head and one tail now? Clearly it is 1/3. If the question said that given the 1st toss is head what is the possibility of one head and one tail then the answer would be 1/2.

Hope this makes it clearer!
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### #22 skale

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Posted 17 July 2007 - 04:07 PM

the point is:
if whether you had a boy or girl fist does not affect what you have later

the choices therefore are
?/boy(50%)
?/girl (50%)

Yes and No.

Yes
if whether you had a boy or girl fist does not affect what you have later

No:
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### #23 unreality

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Posted 17 July 2007 - 10:46 PM

all you ppl that say its 50/50 cuz previous events dont effect the current event are just repeating what i said, but then i saw skale meant "as a whole, looking over the whole thing after two children are born" which it is indeed 1/3 not 1/2, even though the way the puzzle was worded was bad and sugested 1/2
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### #24 willy

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Posted 19 July 2007 - 10:28 PM

you meet a family with one child. what are the odds that child's a girl?

you meet a family with 3 children and are informed that 2 of them are girls. what are the odds the last one's a girl?

you meet a family with 20 children and are informed that 19 of them are girls. what are the odds the last one's a girl?

common sense tells us the answer is 50-50 to all of these. statistical analysis tells us they aren't all equal. i think common sense is the right answer here. all of these say exactly the same thing as my first example. there is one child... odds are 50-50.

i'm familiar with skale's analysis and can confirm that most statisticians would agree. but there is even debate among mathematicians as to what the correct answer is. i personally reject the accepted answer and content this problem is too vague to be solved even by math standards.

the problem is a question of the sampling. does the fact that you know the sex of the first child reduce your sample to just families with one daughter or is the original sample of a totally random population still hold. statistics says to use the original sample giving the 1/3 answer but i say that's wrong. as soon as you know the first child is a girl, your sample changes to only families with at least one girl leaving only 2 possibilities. girl/girl and girl/boy and the 1/2 answer. so you see that even in math circles, this question is still debatable. as i said... i'm going with common sense answer. but technically the answer is 1/3... too bad when math doesn't agree with common sense, isn't it?
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### #25 Garrek99

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Posted 20 July 2007 - 12:38 AM

The distribution of families with 2 kinders is
25% boy-boy
25% girl-girl and the remaining
50% boy-girl
that's understood.

I got confused because I wrongly assumed that we are prognosing what the sex of the second kinder is going to be knowing that the first child is a girl. But, that's not what you were saying at all.
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### #26 Riddari

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Posted 20 July 2007 - 02:31 AM

you meet a family with one child. what are the odds that child's a girl?

you meet a family with 3 children and are informed that 2 of them are girls. what are the odds the last one's a girl?

you meet a family with 20 children and are informed that 19 of them are girls. what are the odds the last one's a girl?

common sense tells us the answer is 50-50 to all of these. statistical analysis tells us they aren't all equal. i think common sense is the right answer here. all of these say exactly the same thing as my first example. there is one child... odds are 50-50.

i'm familiar with skale's analysis and can confirm that most statisticians would agree. but there is even debate among mathematicians as to what the correct answer is. i personally reject the accepted answer and content this problem is too vague to be solved even by math standards.

the problem is a question of the sampling. does the fact that you know the sex of the first child reduce your sample to just families with one daughter or is the original sample of a totally random population still hold. statistics says to use the original sample giving the 1/3 answer but i say that's wrong. as soon as you know the first child is a girl, your sample changes to only families with at least one girl leaving only 2 possibilities. girl/girl and girl/boy and the 1/2 answer. so you see that even in math circles, this question is still debatable. as i said... i'm going with common sense answer. but technically the answer is 1/3... too bad when math doesn't agree with common sense, isn't it?

You guys are reading the question posed by the puzzle incorrectly. The puzzle does not state that the oldest child is a girl and then ask what the probability is for the youngest child to be a girl. If that were the case, 50/50 would be correct. Instead the puzzle states that one of the children is a girl and then asks what the probability is for the other child to also be a girl. Since there are two children with two possible genders, there are four possible combinations. One of those combinations requires that both children be boys, which we know is not possible. So, there are three possible combinations. Only one of these three possibilities includes two girls. Therefore, the probability that the unspecified child is a girl is one in three.

If my math is correct, a family with twenty children would have 2^20 or 1,048,576 possible combinations, which I am not willing to detail out. Knowing that nineteen of those children are girls would eliminate a lot of the possibilities. In fact, there would be only twenty-one remaining possibilities. One where they are all girls and twenty representing a boy in each position. So, there would be a one in twenty-one chance of the unknown child being a girl.

To put it in terms of sampling, your population would change to families that have an oldest girl or a youngest girl, not exclusively. So, it would include all families who have boy/girl, girl/boy and girl/girl, where the oldest child is listed first. It should be obvious, both statistically and intuitively, that the girl/girl families would constitute approximately a third of this population.
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### #27 unreality

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Posted 20 July 2007 - 05:22 PM

The puzzle does not state that the oldest child is a girl and then ask what the probability is for the youngest child to be a girl. If that were the case, 50/50 would be correct. Instead the puzzle states that one of the children is a girl and then asks what the probability is for the other child to also be a girl. Since there are two children with two possible genders, there are four possible combinations. One of those combinations requires that both children be boys, which we know is not possible. So, there are three possible combinations. Only one of these three possibilities includes two girls. Therefore, the probability that the unspecified child is a girl is one in three.

let me quote again the important part of what you said:

the puzzle states that one of the children is a girl and then asks what the probability is for the other child to also be a girl.

then you said there are four possible combinations. YOU ARE WRONG! Yes there are four when looking at both children together. But look at my last quote of you...

for the other child to also be a girl

it doesnt matter what came before. The way the question is worded, we arent looking at as a whole. We're looking at one individual case. It's 1/2

reword the problem and then you will be right with 1/3
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### #28 dsu

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Posted 21 July 2007 - 01:19 AM

Probability does not work that way!

Correct definition:
P(A) = (The Number Of Ways Event 'A' Can Occur) /
(The Total Number Of Possible Outcomes)

Incorrect interpretetion:
P(A) = (The Total Number Of Possible Outcomes)- (The Number Of Ways Event 'A' Has Occured) /
(The Total Number Of Possible Outcomes)

If you toss a coin, and the first two tosses were heads, that does not influence(reduce/increase) the probability of future tosses: it is still 50% or 1/2
[some may argue that the amount of metal on one side is more, and there is greater chance of lighter side to come up]

If you throw a dice, the probability of any number after the 100th throw is still 1/6.
[again some may argue that the center of gravity of a dice is not precisely its center because of the different amount removed from each face]

Similarly in genetic mating, from statistical viewpoint the chance of a boy or a girl each time is 50% or 1/2. Having a boy or girl once does not reduce or increase the chance the next time [in fact some may argue some families have only girls and some have only boys].

The sum and substance is - if we see the first sibling, that in no way affects the probability of seeing the sex of the next sibling.
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### #29 unreality

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Posted 21 July 2007 - 05:18 PM

exactly what i said on the first page! its 1/2! I'm just saying, they could reword the problem to be looking in retrospect, saying "one of two children is a girl, what is the prob they are both girls?" would be a good rewording of it, where the answer is 1/3

the way they worded it, the answer is clearly 1/2

if they said this instead: "one of two children is a girl, what is the prob they are both girls?" the answer for that is 1/3
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### #30 Martini

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Posted 22 July 2007 - 04:41 AM

exactly what i said on the first page! its 1/2! I'm just saying, they could reword the problem to be looking in retrospect, saying "one of two children is a girl, what is the prob they are both girls?" would be a good rewording of it, where the answer is 1/3

the way they worded it, the answer is clearly 1/2

if they said this instead: "one of two children is a girl, what is the prob they are both girls?" the answer for that is 1/3

There is nothing wrong with the wording in the OP.

They have two kids, one of them is a girl, what is the probability that the other kid is also a girl.

Asking "what is the prob they are both girls?" is effectively the same as asking "what is the probability that the other kid is also a girl?"

Either way, the probability is 1/3.
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