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Twins puzzle

Helen7

Helen7

LostCousins Superstar
I'm no statistician, but in response to the twin sheep puzzle, I would think the answer is two-thirds or 2 in 3 that the other twin is female.

My reasoning is this:
There are 4 possible combinations for the M/F sexes of the twins:
M-M
M-F
F-M
F-F

So without the genetic test, the chances of one male and one female is 50%.

However, the genetic test removes one of the possibilities, i.e. the F-F, leaving just 3 possible combinations, two of which include a female. So the odds are now 2/3 in favour of the second twin being female.

Is this correct?
 
I'm no statistician either and I was going to suggest that the answer is dependent on whether the twins are identical or fraternal.

However, I have a feeling that identical twins in sheep are pretty rare, so on the assumption that the twins are almost certainly fraternal, I think the sex of the second twin will be unaffected by the sex of the first.

That would mean there is a 50% chance that the second twin is female.

That figure would probably need to be reduced slightly to take account of the small chance of the twins being identical.
 
Helen7 said:
However, the genetic test removes one of the possibilities, i.e. the F-F, leaving just 3 possible combinations, two of which include a female. So the odds are now 2/3 in favour of the second twin being female.

Is this correct?
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I came to the same conclusion before looking on this forum, with the same reasoning.
 
Pauline said:
However, I have a feeling that identical twins in sheep are pretty rare, so on the assumption that the twins are almost certainly fraternal, I think the sex of the second twin will be unaffected by the sex of the first.
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Ah, but which is the first twin and which is the second?
 
Pauline said:
I was going to suggest that the answer is dependent on whether the twins are identical or fraternal.
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Well-spotted - the analysis in New Scientist didn't take this into account, and whilst I wrote in to remind them of this possibility, they haven't published the letter (yet).
 
peter said:
Ah, but which is the first twin and which is the second?
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That was just my shorthand - I wasn't putting them in order of conception or anything. I probably shouldn't have used the words "first" and "second" at all.
 
Pauline said:
That was just my shorthand - I wasn't putting them in order of conception or anything. I probably shouldn't have used the words "first" and "second" at all.
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Similarly my response was my shorthand pointing you in the direction of the flaw in your reasoning. If the order of birth made no difference then there would have been only 3 possibilities at the outset, not 4 as listed by Helen7.
 
peter said:
Similarly my response was my shorthand pointing you in the direction of the flaw in your reasoning. If the order of birth made no difference then there would have been only 3 possibilities at the outset, not 4 as listed by Helen7.
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If you take any random sheep having twins, then she may be carrying two male lambs, two female or one of each.

However, we are not talking about a random sheep here, but one where at least one of her twins is known to be male.

My thinking was, does knowing that one of the twins is male tell us anything about the other twin, assuming fraternal twins? I would say not - each twin has a 50% chance of being female regardless of what the other twin happens to be.
 
Pauline said:
each twin has a 50% chance of being female regardless of what the other twin happens to be
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That's certainly true at the start. But then you have to take into account the new information.

The DNA test eliminates F-F. The question is, how are the other possibilities affected by that new information?

Clearly at the outset there was a 50% chance of the twins being the same sex; if we accept your solution then there's still a 50% chance of the twins being the same gender even though F-F has been eliminated. Does that seem reasonable to you?
 
Label the two uteruses as A and B. In principle either could be male or female so the possibilities are MM, MF, FM or FF. However we can eliminate FF from the blood test. Of the 3 remaining 2 include a F so the odds are 2 in 3.
 
peter said:
Clearly at the outset there was a 50% chance of the twins being the same sex; if we accept your solution then there's still a 50% chance of the twins being the same gender even though F-F has been eliminated. Does that seem reasonable to you?
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When you put it like that, no!
 
Let's call the sheep "Sheep 1" and "Sheep 2". Looking at the 4 possibilities put forward by Helen7 (see the first reply):
  • If sheep 1 was tested, we can eliminate F-M and F-F as Sheep 1 has been tested as being male.
  • If sheep 2 was tested, we can eliminate M-F and F-F as Sheep 2 has been tested as being male.
  • In both cases, we are left with two possibilities, with the other sheep having a 50% chance of being female. This does not of course take into account the possibility of both sheep being more likely than not of being the same sex.
 
webwiz said:
The maths behind this is the same as the famous Monty Hall problem.
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The Monty Hall problem was featured in the newsletter on 2013 - the problem and solution were stated here.

Mathematically they're very similar problems, but there are some added complications to the Monty Hall problem (eg does the host always make the same offer).
 
At the risk of digging myself deeper into a hole here, I think what confused me was, a while back, the question was posed that if someone was expecting their second child and already had a daughter, what was the likelihood of the second child being a boy (excluding medical & genetic factors).

I think - hope - I am right in saying that a subtle difference with this question is that we not only know that one of the two children is a girl, we also which of them is a girl.
 
Pauline said:
At the risk of digging myself deeper into a hole here, I think what confused me was, a while back, the question was posed that if someone was expecting their second child and already had a daughter, what was the likelihood of the second child being a boy (excluding medical & genetic factors).

I think - hope - I am right in saying that a subtle difference with this question is that we not only know that one of the two children is a girl, we also which of them is a girl.
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We know that at least one is male - nothing more.
 
peter said:
We know that at least one is male - nothing more.
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Yes, I wasn't questioning that, just commenting on getting confused. :confused:

I had statistics buzzing around my head last night which hindered my sleep somewhat. I tried counting sheep but that didn't really help! ;)
 
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