Boy or girl paradox

[Huan Dao]

Question: You met a high school friend in a shop, he told you that he had two children and at least one boy. What is the probability that the other child is a boy.
Assumption:
1. Boys to girls ratio is 50:50
2. The sex of each child is independent of the sex of the other.

 

[AlexandreH]

P(child2=boy|child1=boy)
= P(child2=boy AND child1=boy)/P(child1=boy)
= P(child2=boy)=1/2
I am sure I am missing something here?

 

[ANielsen]

1/3 ?

More interesting. What if he told you that the boy was born on a sunday?

 

[Huan Dao]

@AlexandreH P(child1=boy) should be 3/4 because the question did not specify whether the boy is older or younger child. However, with P(child1=boy)=3/4 your answer is not correct either.

@AN 1/3 is wrong

 

[ANielsen]

Really? Maybe i should take a statistics course
One can get two boys, a boy and then a girl, the other way around or two girls, each of these outcomes are equally likely under your assumption. He told you that one was a boy, and your left with the posibilities bb bg gb, so the probabilty of two boys is 1/3?

 

[AlexandreH]

I guess there are 2 answers to this problem depending on how you understand it. The answers are 1/3 and 1/2.
http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

 

[pruse]

@AlexandreH P(child1=boy) should be 3/4 because the question did not specify whether the boy is older or younger child. However, with P(child1=boy)=3/4 your answer is not correct either.

@AN 1/3 is wrong

No, AN is right, and from the way the question is phrased, there is only one correct answer (1/3! not 1/2!)

for the Sunday question, if the first child is a Sunday boy, there are 14 possibilities for the second child. if the second child is a Sunday boy there are again 14 possibilities for the first child. so there are 14+14-1=27 ways to have at least one Sunday boy. Out of these, there are 7+7=14 ways for the other child to be a girl. so the answer is (27-14)/27=13/27.

 

[AlexandreH]

No, AN is right, and from the way the question is phrased, there is only one correct answer (1/3! not 1/2!)

for the Sunday question, if the first child is a Sunday boy, there are 14 possibilities for the second child. if the second child is a Sunday boy there are again 14 possibilities for the first child. so there are 14+14-1=27 ways to have at least one Sunday boy. Out of these, there are 7+7=14 ways for the other child to be a girl. so the answer is (27-14)/27=13/27.

The way it is phrased could go either way. It is a Paradox, no right answer. I did not know about it but can see how u can solve it in 2 diff ways. The Wiki link is quite useful.

 

[Joe Grippi]

http://www.maa.org/devlin/devlin_04_10.html

 

[pruse]

The way it is phrased could go either way. It is a Paradox, no right answer. I did not know about it but can see how u can solve it in 2 diff ways. The Wiki link is quite useful.

yes, and if you read the wiki link, you'll see that, given the phrasing, the correct answer is 1/3. the link even gives some empirical proof of it.

 

[AlexandreH]

Come on Peter. This is as confusing and ambiguous as it can be.
From Wiki: "Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"
Question asked here: "You met a high school friend in a shop, he told you that he had two children and at least one boy. What is the probability that the other child is a boy."
Now, Don't tell me the first one is confusing and the second one gives you 1/3.
The link provided by Joe Grippi is more convincing, but still, even Mathematicians don't agree on this problem.
The is no empirical proof. That paragraph was even less concluding!!!


Actually, if you look at the way it was asked here ("You met a high school friend in a shop, he told you that he had two children and at least one boy. What is the probability that the other child is a boy."), it is similar to "'I have two children and at least one of them is a boy.' Given this information, what is the probability that the other child is a boy?" found in the "Scientific investigation" paragraph in the wiki link. This is the most confusing version and the one that leads to an answer of 1/2 as explained here:

• "Mr. Smith says: 'I have two children and at least one of them is a boy.' Given this information, what is the probability that the other child is a boy?"
• "Mr. Smith says: 'I have two children and it is not the case that they are both girls.' Given this information, what is the probability that both children are boys?"

The authors argue that the first formulation gives the reader the mistaken impression that there are two possible outcomes for the "other child", whereas the second formulation gives the reader the impression that there are four possible outcomes, of which one has been rejected (resulting in 1/3 being the probability of both children being boys, as there are 3 remaining possible outcomes, only one of which is that both of the children are boys). The study found that 85% of participants answered 1/2 for the first formulation, while only 39% responded that way to the second formulation. The authors argued that the reason people respond differently to this question is because of the use of naive heuristics that fail to properly define the number of possible outcomes.

 

[FedericoM]

I guess there are 2 answers to this problem depending on how you understand it. The answers are 1/3 and 1/2.
http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

This was one of the questions I had to solve in my final high-school exam!

 

[Tsotne]

Heey guys i cannot belive eyes. What is 1/3? Why such philosophy? That's it:

P(child2 = boy | child1 = boy) = P(child2 = boy) since the given information doesnt at all have any effect on the probability of the second child being girl or boy. They are independent from the math point of view. So the probability of both being boy is the intersection of the individual probabilities that each of them are boy. So the probability that the first child is boy given that he has already been born and is boy is 1. Second child left. The probability of the second child being boy is 0.5. So 0.5*1 = 0.5.

This thread is DEAD.

 

[Tsotne]

It can be more interesting if the dependency occurs. I'm not keen on medicine and biology but if the biologists tell us that the event of second child being male is somehow depended on what the first was, then we'd calculate other way.

 

[Tsotne]

I misunderstood. The previous posts deal with different question. If she told that the first was already boy. Here the question is asked incorrectly.

I guess there are 2 answers to this problem depending on how you understand it

That's right. Look how everything develops. We can view this problem as the binomial distribution with 0,1,2 bernulli outcomes, independent of each other with the same probability of success(boy) equal to 0.5. Now, if we construct the probability distribution function we see this:


x f(x)
0 0.25
1 0.5
2 0.25
Binomdist(x,2,0.5,0) excel formula.

So that old friend told us that she had at least 1 boy out of 2 children. Since these events (0 boy, 1 boy, 2 boys) are mutually exclusive and collectively exhaustive we can view that as the union of 1 boy and 2 boys is the sum of their probabilities. So 0.5+0.25=0.75.

But here's why I think that the question is asked incorrectly:

What is the probability that the other child is a boy.

Before we hit "other child", which is non-other?
Since now I have no clue how to go on. I can calculate probability if the question is cleared.

 

[Tsotne]

Anybody please make it clear who is the "other child"

 

[pruse]

The question is not ambiguous at all. The answer is clearly 1/3.

 

[Tsotne]

The question is not ambiguous at all. The answer is clearly 1/3.

I don't say the answer is not 1/3. I'm just asking who is "other child". Just didn't understand the question. Ok tell me how you calculated that 1/3 and I'll guess myself what the question is asking.

 

[pruse]

I don't say the answer is not 1/3. I'm just asking who is "other child". Just didn't understand the question. Ok tell me how you calculated that 1/3 and I'll guess myself what the question is asking.

There are two children, and you know one of the children is a boy. The "other" child is the child we don't know have information about. The problem can be rephrased as, Find the probability that both children are boys, given that you know one of them is a boy.

 

[Tsotne]

Maaaan thats 1/2. Definitely with no doubt. The explanation is here in my older post:

Heey guys i cannot belive eyes. What is 1/3? Why such philosophy? That's it:

P(child2 = boy | child1 = boy) = P(child2 = boy) since the given information doesnt at all have any effect on the probability of the second child being girl or boy. They are independent from the math point of view. So the probability of both being boy is the intersection of the individual probabilities that each of them are boy. So the probability that the first child is boy given that he has already been born and is boy is 1. Second child left. The probability of the second child being boy is 0.5. So 0.5*1 = 0.5.

This thread is DEAD.

BTW I really wonder how did you get 1/3?