Twins puzzle

[Helen7]

I've just put this puzzle to my son, who is a mathematician, and his first response was "it depends on the relative likelihood of fraternal versus identical twins". Then when I told him that identical twins are rare in sheep (<1% of twin births), he immediately came back with the answer 2/3. No hesitation. I'm inclined to believe him.

If the first lamb is male then there are only 2 possible outcomes, MM and MF. If the second lamb is male then there are also only two possible outcomes, MM and FM. Hence the probability of one lamb being female is 2/4, ie 1/2.

When you say 2/4, what are your four possible outcomes? As far as I can see, amalgamating the possibilities from the two scenarios (first lamb male, second lamb male) gives only 3 options: MM, MF, FM, hence probability of the other twin being female is 2/3 not 2/4.

 

[Pauline]

Initially, I reasoned along similar lines to Helen7 in post #1 but then thought that there was possibly more to be considered and decided that Pauline in post #2 was more correct.

I'm not usually averse to people agreeing with me, but on this occasion I've already acknowledged I was wrong in #2 - and thus, by implication, that Helen was right in #1.

 

[webwiz]

What's wrong with this analysis?

Imagine 1000 sheep all carrying twins with equal chance of gender throughout so there are 1000 male and 1000 female foetuses. They are all tested and as expected 75% carry male DNA. None of the 25% FF will go on to have a male so the 1000 male foetuses will be among the 1500 in the other groups leaving 500 female. So the probability of any one being female is not 50% or 66% but only 33%.

 

[Helen7]

the 1000 male foetuses will be among the 1500 in the other groups leaving 500 female.

But of those 1500 foetuses in the other groups, 750 have been shown to be male by the genetic test, so that leaves only 750 of unknown sex, of which 500 are female, so chances of the second twin being female is 500/750 or 2/3.

 

[webwiz]

But of those 1500 foetuses in the other groups, 750 have been shown to be male by the genetic test, so that leaves only 750 of unknown sex, of which 500 are female, so chances of the second twin being female is 500/750 or 2/3.

Yes that's one way of putting it, but I prefer to calculate the chance of both being male, which is clearly 25% of the original 1000 or one third of the 750, so the chance of one being female is the complementary 2/3.

 

[Bryman]

When you say 2/4, what are your four possible outcomes? As far as I can see, amalgamating the possibilities from the two scenarios (first lamb male, second lamb male) gives only 3 options: MM, MF, FM, hence probability of the other twin being female is 2/3 not 2/4.

The four possible outcomes are MM and MF from consideration of the first lamb being male, together with MM and FM from consideration of the second lamb being male. Although there are two similar instances of MM, they are each obtained from different possibilities. I don't believe that there is any valid reason why one of them should be arbitrarily ignored in order to only allow unique situations to be included in the calculation. If I am wrong in that belief then the solution would appear to be 2/3 instead of 1/2.

 

[PhilGee]

I must admit to being confused I am trying to understand why there are four possibilities - FF FM MF MM. That indicates that order is significant, which is surely not the case. If you now eliminate FF as it is known one is M, that leaves two possibilities.

I'm open to corrections (mathematically accurate) explained in simple terms.

 

[webwiz]

I must admit to being confused I am trying to understand why there are four possibilities - FF FM MF MM. That indicates that order is significant, which is surely not the case. If you now eliminate FF as it is known one is M, that leaves two possibilities.

I'm open to corrections (mathematically accurate) explained in simple terms.

You have to somehow identify the two foetuses. It can be arbitrary or the heaviest or the first conceived or any way you want. Do you agree that (subject to gender likelihood equality, and ignoring biological distractions) that half of a population of twinning sheep will have two of the same gender and the other half one of each? If you treat MF and FM as the same and there are only 3 possibilities then 2/3 of twins would be the same gender.

 

[Pauline]

I must admit to being confused I am trying to understand why there are four possibilities - FF FM MF MM

You can, if you prefer, think of it as 3 possibilities. Take any random set of non-identical twins and there will be a 25% chance of both being male, a 25 % chance of both being female and a 50 % chance of one of each.

 

[peter]

But that is exactly the problem with your analysis. Why have you eliminated only FF? One of the MF/FM possibilities should also be eliminated, depending on which of the two lambs is assumed to be male. We don't know which of the 2 lambs is male so have to consider each possibility in turn. If the first lamb is male then there are only 2 possible outcomes, MM and MF. If the second lamb is male then there are also only two possible outcomes, MM and FM. Hence the probability of one lamb being female is 2/4, ie 1/2.

Any Monte Carlo simulation that you conduct will only reinforce your incorrect assumption and lead you to believe that the probability is 2/3. Using my extended analysis, a Monte Carlo simulation will indicate a probability of 1/2. It all depends on the conditions defined for the simulation.

At the beginning there are 4 equally likely possibilities, thus the chance that the twins are the same gender is 1/2.

The implication of your line of reasoning is that even though the possibility of both twins being female has been eliminated by the DNA test, the chance of both twins being the same gender is still 1/2. Do you think that is correct?

 

[peter]

I am trying to understand why there are four possibilities - FF FM MF MM. That indicates that order is significant, which is surely not the case.

Imagine you're tossing a coin which comes up heads or tails with equal frequency (a 'fair coin'). Tossing the coin twice in succession is equivalent to the problem under discussion, and clearly there are 4 possible outcomes which are equally likely.

If you tossed two identical coins simultaneously it wouldn't change the odds of the different outcomes, but unless you were watching closely you might not know which coin was which.

 

[TerryM]

I will acknowledge that I was in error with the door puzzle, after viewing that explanation link I have realised my error. When the host removes one door option it is not a random selection but instead a deliberate selection of one outcome which is why it then influences further outcomes down the track. In the maths problems in my boys high school classes the wording is always very specific to note that there is no selection in the first choice and that is why it can be ignored for further calculations of odds.

However in the case of the twin sheep I am still struggling to see where the selection pressure is. Knowing one twin is male surely takes both options off the table for that lamb as it is a known outcome. Leaving one lamb as an unknown. It is quite a different problem to the door puzzle where their is three 'lambs' of known sex and one is removed in a selective way leaving you to guess where the 'singleton sex' might be.

 

[Pauline]

Knowing one twin is male surely takes both options off the table for that lamb as it is a known outcome

Which lamb is “that lamb”? The important thing here is that while we know that one of the twins is male, we don’t know which of the twins it is.

 

[peter]

However in the case of the twin sheep I am still struggling to see where the selection pressure is

How would you counter the argument in my post #50?

 

[peter]

Perhaps the reason that this puzzle is so difficult is that we know that there's a 50-50 chance of a lamb being female, so it goes against our intuition to be told that the chance of a lamb being female has apparently increased to 2/3 as a result of discovering that at least one of the twins is male.

However it becomes less counter-intuitive when we realise that at the beginning the chance of at least one of the lambs being female was not 1/2 but 3/4.

 

[Bob Spiers]

When the host removes one door option it is not a random selection but instead a deliberate selection of one outcome which is why it then influences further outcomes down the track. In the maths problems in my boys high school classes the wording is always very specific to note that there is no selection in the first choice and that is why it can be ignored for further calculations of odds.

I cannot see how a deliberate selection -removing a 'false' remaining door -as opposed to a random 'true/false' selection, makes any difference to the outcome, of being left with a 50/50 choice. Obviously if a random selection happened to remove a 'true' door you would still be left with two choices, the fact that both would be 'wrong' does not change things, nor the odds.

 

[Pauline]

Some of you might think that it doesn't matter, it's just a puzzle. But what if you make the same errors in reasoning when you're researching your tree? We discover new information all the time - we need to be able to assess how that it affects any assumptions or estimates that we made previously.

I was wondering if you might be able to expand on this a bit, please, maybe with an example or two. I don't consciously think about probabilities when doing my tree except in a very broad sense.

New information might enable me add to add something new into my tree, or to further confirm what I already knew but, as far as my direct line is concerned, I try and avoid making assumptions in the first place.

 

[webwiz]

There is no certainty in Family History (unless you count negative certainty). Every time we add a person to our tree, starting with out mother and father, we are making an assumption. As we go farther back our assumptions tend to be less secure, and we are relying on probabilities. So some sense of quantifying probability is required, and as the two examples discussed in this thread demonstrate, this can be quite tricky and controversial.

 

[Pauline]

As we go farther back our assumptions tend to be less secure, and we are relying on probabilities.

Only in a very broad sense, surely, in that after weighing up all the available evidence, we may make a decision about whether something is, say, pretty certain, probable, possible or very unlikely.

 

[peter]

I was wondering if you might be able to expand on this a bit, please, maybe with an example or two. I don't consciously think about probabilities when doing my tree except in a very broad sense.

New information might enable me add to add something new into my tree, or to further confirm what I already knew but, as far as my direct line is concerned, I try and avoid making assumptions in the first place.

There will be some examples in forthcoming newsletter articles.

Only in a very broad sense, surely, in that after weighing up all the available evidence, we may make a decision about whether something is, say, pretty certain, probable, possible or very unlikely.

We might not put percentages on it, but we do need to be able to adjust our estimates appropriately as new information comes to light. For example, if there are three candidates for our ancestor's baptism we might initially regard them as equally likely, but subsequently tweak them as we learn more.

Equally important is recognising when the odds don't change as the result of new information becoming available. This is the key to the doors problem, where the chances of our initial selection being correct don't change when the host opens another door (provided the door he opens isn't the one with the prize).

Taking the three baptisms example, how do the odds change if looking through the burial register we find that one - or even two - of the candidates died as infants?