345 replies to this topic

[bonanova]

"Other" answers to "One." it does not answer to "At least one."
The OP says "One," so it appropriately asks about the "other."

But be careful not to identify "One" with a specific one.
That leads to 1/2 unambiguously, but the OP does not support that interpretation

[syonidv]

That leads to 1/2 unambiguously, but the OP does not support that interpretation

I believe the mere presence of "other" in the OP implies that our given information pertains specifically to one individual (not two collectively) and the question asks us to infer information about the "other" individual.

At its heart, the 1/3-ist interpretation is that the given information pertains to the joint distribution of two individuals, and the solution is an inference that can be made about the conditional joint distribution (i.e. the fact that in the FF case, we may conveniently neglect the issue of selecting an "other" and conclude "female" since both siblings are female).

But perhaps this speaks to the point of many posts in this forum, which is that the problem is poorly stated. The answer depends on what specifically is meant by "two kids; one girl"--how that information is obtained.

Thanks to all for the discussion, at any rate.

Edited by syonidv, 18 July 2012 - 04:07 PM.

[bonanova]

I believe the mere presence of "other" in the OP implies that our given information pertains specifically to one individual (not two collectively) and the question asks us to infer information about the "other" individual.

At its heart, the 1/3-ist interpretation is that the given information pertains to the joint distribution of two individuals, and the solution is an inference that can be made about the conditional joint distribution (i.e. the fact that in the FF case, we may conveniently neglect the issue of selecting an "other" and conclude "female" since both siblings are female).

But perhaps this speaks to the point of many posts in this forum, which is that the problem is poorly stated. The answer depends on what specifically is meant by "two kids; one girl"--how that information is obtained.

Thanks to all for the discussion, at any rate.

Well said. If the OP had said "It's not the case that they are both boys. What is the probability that they are both girls?" this forum would have been much shorter.

Thanks to you, for posting your thoughts clearly and cogently.

[Kikacat123]

The probability that they are both girls is 1/2. We know that one child is a girl. This does not affect the probability of what the other child is. If you are equally likely to have a boy or a girl, having another child does not make any change to that. In fact, there are only 3 possible combinations of gender, two boys, two girls, and one of each. Since two boys are not possible, there is a 1/2 probability of the other child being a girl.

[ParaLogic]

At first, I thought this 35 page long argument was ridiculous, but after reading bona/syon's posts above, I can understand some of the trouble surrounding the ambiguity of the question. This is the way I see it: 

 

There are four possibilities for boy/girl combinations, as mentioned several times by now (I imagine):

 

BB

BG

GB

GG

 

Note that this categorization does take into account the order of the children. We are shown that one of the children is a girl, which of course eliminates the first combination.This is the point where people jump to the answer of 1/3, since there only seem to be three combinations remaining. However, as I stated before, order does matter if you look at the problem this way. Therefore, there are still four possibilities:

 

G1B2

G2B1

G1G2

G2G1

 

Where the number indicates order. The girl we see can be either the elder (G1) or younger (G2) sister, so we must look at both possibilities equally. Thus, the probability stands at 1/2, regardless of the order of birth.

 

Another way to look at this is to completely disregard order in all of the cases. The resulting combinations are:

 

BB

GB (same as BG!)

GG

 

When BB is eliminated, the probability of GG remains at 1/2.

Edited by ParaLogic, 14 April 2013 - 09:10 PM.

[bonanova]

The probability that they are both girls is 1/2. We know that one child is a girl. This does not affect the probability of what the other child is. If you are equally likely to have a boy or a girl, having another child does not make any change to that. In fact, there are only 3 possible combinations of gender, two boys, two girls, and one of each. Since two boys are not possible, there is a 1/2 probability of the other child being a girl.

1. If OP is taken to identify "one" of the two, and to say that "one" is a girl, the probability is indeed 1/2 that the "other" is, too.
    
2. If OP is taken to say it is not the case that they are both boys, then one can deduce a probability of 1/3, although it is not necessary to conclude 1/3 since our fact reporter's algorithm has not been disclosed.
    
3. It is incorrect to say there are 3 combinations and one is removed so p=1/2. That reasoning ignores the equal-likelihood requirement: I buy a lottery ticket, and there are only two possibilities: winning and losing. Therefore my chances of winning are ...
    
4. Before adding to this thread, read through some of the posts first.
    
5. Additional, thoughtful comments are always appropriate.