MegaSack DRAW - This year's winner is user - rgwb
We will be in touch
Hehehe... does anyone remember this, which caused some debate I seem to recall... ;o)
If I have two children, one of which is a girl, what is the probability that I have two girls?
Well, a new twist seems to have appeared on this one. A person asked:
If I have two children, one being a boy born on a Tuesday, what is the probability that I have two boys?
They then proceeded to say
The first thing you think is 'What has Tuesday got to do with it?' Well, it has everything to do with it.
So all you great minds out there... what do we think?
Still 50%. Tuesday is irrelevant. All IMHO.
it is either 33% if you wish to use the scientific methodology to get your answer - this more accurately reflects reality
or
50% if you wish to use binary deductive logic as it is either a boy or a girl and that will be right 100% of the time
I fail to see what Tuesday has to do with it though i understand the maths behind it and the result is 13/27.
I very much doubt its empirically substantiated -it actually happens in the real world - as essentially in scenario one you have a boy born on a certain day [ you have just not stated the day] and may just be a mathematical artefact.
Essentially numbers and sytems make approximations of relaity that ,ay or may not be accurate - we can actually test the Tuesday thing has anyone?
I seem to remember the the sex of a child can be influenced by the amount of time the sperm is "hanging around" before being used. Something to do with male sperm lasting longer than female sperm.
Or is it the other way around? Can't find the info I need to remind me...
The original puzzle for those that missed it:
www.singletrackworld.com/forum/topic/the-boy-girl-puzzle
And [b]please, for the love of Kylie, read it before saying "It's 50%".[/b] It's not. And it's not some weird statistics anomaly - it is the actual truth. Fact.
Anyway, this new twist, hmmm, I can't get the relevance of the Tuesday, so (assuming we're in a world where the chance of a boy or girl is exactly 50%) I'll stick with the answer of 33% but I'm listening with interest...
Ah wait, does "[u]one[/u] being born on a Tuesday" imply that the other one wasn't?
grahams - indeed, comes down to semantics
or does it ?
If I tell you that I wasn't born on a Tuesday, does that mean threre's a 1/7 chance I don't exist or a 1/13 chance that I'm female
(looks at chest) hmmmmm, you may be onto something
Retracts earlier post 😳
33%, again IMHO! Still don't see what Tuesday has to do with it.
It adds the variable of tuesday and this alters the probability- see below. However, unlike with knowing the gender- which eradicats the female/female option- I cannot see how this actually alter the actual likelyhodd of anything changing in the real world if we empirically measued data for all males born on a Tuesday. In the first scenario the boy has to be born on some day anyway so nothing really has changed. It is interesting but not unlikely to be matched by observation IMHO
Here is the maths for interest
Let's list the equally likely possibilities of children, together with the days of the week they are born in. Let's call a boy born on a Tuesday a BTu. Our possible situations are:?When the first child is a BTu and the second is a girl born on any day of the week: there are seven different possibilities.
?When the first child is a girl born on any day of the week and the second is a BTu: again, there are seven different possibilities.
?When the first child is a BTu and the second is a boy born on any day of the week: again there are seven different possibilities.
?Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a BTu – and this is where it gets interesting. There are seven different possibilities here too, but one of them – when both boys are born on a Tuesday – has already been counted when we considered the first to be a BTu and the second on any day of the week. So, since we are counting equally likely possibilities, we can only find an extra six possibilities here.Summing up the totals, there are 7 + 7 + 7 + 6 = 27 different equally likely combinations of children with specified gender and birth day, and 13 of these combinations are two boys. So the answer is 13/27, which is very different from 1/3.
It seems remarkable that the probability of having two boys changes from 1/3 to 13/27 when the birth day of one boy is stated – yet it does, and it's quite a generous difference at that. In fact, if you repeat the question but specify a trait rarer than 1/7 (the chance of being born on a Tuesday), the closer the probability will approach 1/2.
...if we empirically measued data for all males born on a Tuesday...
I think that's the point though - the situation is asking what you can infer from what the statements tell you about ONE existant family
If the sentence is " and ONLY one of them is a boy born on Tuesday ", then the increased likelihood of the other child being female is probably true.
If the (unspoken) 4th sentence COULD be "and the other one is also", then you're back on the simple maths
(I do prefer the moobs version myself though) 😉 😳
Oh the f@@k dear. All you would-be mathematicians are missing a crucial part of Quantum Analysis. Combinations and permutations are two different concepts and will give wildly different answers.
It's 50%.
Probability is easy to understand and is only made to seem complicated by people who want to seem clever.
what you can infer from what the statements tell you about ONE existant family
Correct but if this inference is wrong , that is not matched by actually observing this in the real world, then the inference is empirically inoorrect. The variable does not alter the actual occurence IMHO and therefore the inference is wrong. Nothing has changed to increase the chance of male child you will still observe 33% nor 13/27.
An interesting case where the probability calculation is correct but the answer is wrong ❓
Moobs? No knowledge of this sorry.
OK genghis,
I've got two coins on my desk (UK and can only be 1p 2p 5p 10p 20p 50p or £1)
ONLY one of them is a 1p piece showing tails (but the other could be a 1p showing heads)
You have to bet your house/life on whether the other one is heads or tails
Which would you pick ?
junkyard, I think it's too small a sample to expect to apply population averages
you know...some people unkindly say that the people on this site are geeks.
Who'd have thunk it.
scaredypants - I would choose whichever one took my fancy. I get the whole probability thing. But the coin has already landed, only has two sides and is a separate entity and I prefer logic to probability.
OK Scaredypants,
Assuming that you've already flipped the 1p (resulting in Tails), And assuming that there may or may not be another 1p coin (which you haven't specified) the answer is 50%. The next coin will land on one side or the other. The landing of the previous coin will not affect the outcome.
(I don't think it's 1/3 "normally" though - seems to me that we can double the frequency of Boy & Boy just by naming them; so funkynick eldest & junkyard youngest but also vice versa)
I think I've got this right...
I have two children, one is a boy, what is the probability the other is = 33%
I have wo children, the first is a boy, what is the probability the other is = 50%
I have two children, one is a boy, what is the probability the other is = 33%
If you have it right, I haven't:
so he either has a younger sibling - what sex? (B or G)
or an older sibling - what sex? (B or G)
or, if ONE is a boy, zero%
The only combinations of children we can have with two are : M M. M F. F F. All of which are equally likely 25%. The question states that one child is female. We cannot get M M so we are left with three options, FF, F M , M F - or 33% chance of two females. This is what actually happens in the real world.
Again as I have said saying 50% is clearly correct we only hve two choices male or female and each chance is 50% IF THAT IS ALL YOU KNOW or you want to use dedcuative reasoning only. However as the maths above shows you will actually have F F in 33% of all scenarios with one female as 25% of scenarios ( M M ) cannot occur and are not in your sample.
Assuming that you've already flipped the 1p (resulting in Tails), And assuming that there may or may not be another 1p coin (which you haven't specified) the answer is 50%. The next coin will land on one side or the other. The landing of the previous coin will not affect the outcome.
You are correct in that example but if the two coins are already flipped and you are told one is tails the odds of the other being a tail is not 50 /50. If you want to do this with flipped coins and we flip two get at keast one tail and you call heads and I call tails for the other one . I will do it for £100 a throw for 1000 times and you can see if it was 50/ 50 😆 we throw 25% of coins away and this alters the odds. One result does not cause the other though.
scaredypants - Member
junkyard, I think it's too small a sample to expect to apply population averages
It is a probability so we would need to do it for a number of measures to be certain of it's accuracy. We could get any combination no matter how unlikely on one measure - the odds being 1/27.
You either get this or you dont it is counter intuiative but is is correct. I undersatnd why people say 50/50but itis not correct
igm same question different words 33%. If you have two boys both the first and the second are boys
when you said that stuff about empirically measuring data for boys born on a Tuesday you were using it to make the argument thatWe could get any combination no matter how unlikely on one measure - the odds being 1/27
not sure you can have it both ways 😉I cannot see how this actually alter the actual likelyhodd of anything changing in the real world
IMO, if we're identifying a child (lets say boy a), we have to incorporate the opportunity for that child to be eldest or youngest of a pair.The only combinations of children we can have with two are : M M. M F. F F. All of which are equally likely 25%
4 possibilities Ba:Bb Bb:Ba Ba:G G:Ba 50:50
I knew this would happen.
genghispod: The original answer to the question IS 33%. There is no doubt about this. It is not some weird statistical trick - it's just what actually happens in actual reality. If you doubt this then go back and read the original thread. I even ended up doing a nice google spreadsheet that illustrated it happening right in front of your eyes.
Now, on this Tuesday thing, can the OP confirm if he means ONLY the boy mentioned was born on a Tuesday?
Most babies are born on Tuesday, why? Because doctors done't like working weekends so the c-sections are scheduled after = Tuesday peak. Relevance / c-sec = twins.
Intuitively I'm guessing (because thinking is hard) that this takes us back to 50% (ironically for those that do not understand the 33% thing) because there is an insinuation both were born simultaneously, or was in effect one event?
Or something, maybe,
I even ended up doing a nice google spreadsheet that illustrated it happening right in front of your eyes
THAT, sir, is an outrage - this is from-the-hip pseudomaths, not spreadsheets !
dunno if you were here then, but last time I got involved in one of these (oddly, turned out I was wrong 😉 ) I played a real game on stw for money with a clever man who knew he would win. Much more exciting that way !
(Now I can see that there are twice as many 2-kid families with one of each as there are with 2 boys, but STILL I can't go for the 33% on this 😳 )
4 possibilities Ba:Bb Bb:Ba Ba:G G:Ba 50:50
like the argumene bit in the example cited this si not relevant but I do like it as an argument..interesting point to argue but reality is not 50/50.
(Now I can see that there are twice as many 2-kid families with one of each as there are with 2 boys, but STILL I can't go for the 33% on this 😳
Theres the rub.
Junkyard - Member
igm same question different words 33%.
Not so Junkyard
4 posibilities
MM
MF
FM
FF
Now if one is a boy, you exclude FF and the chance of two boys is 33%
But if the first is a boy, you exclude both FF and FM, leaving MM and MF, so the chance of two boys is 50%.
Always assuming that certain families don't just tend towards boys or girls (and there is evidence that they do I believe).
Would you disagree?
All the calculations are based round a male:female ratio of 50:50.
Something I didn't know until I just googled it - apparently the natural male:female birth ratio in humans is not 1:1 - it is something like 1.05:1. I think this is to offset the fact that the death rate of young males is higher than that of young females and the excess males die off giving us a 1:1 ratio at breeding age. As the question is based around [i]children [/i]there will be more boys than girls in that age group. Therefore we'll need to tweak the calculated result (if the calculation is based on a 50/50 male female split)!
GrahamS is still right though 🙂
Yeah I suspect occurences of paternal vs maternal twins also messes it up. But i did say in the original that for the sake of argument it was a 'perfect' 50:50 world.
Surely a perfect world would be 9 women for every man.
Those arguing for the 50% might just as well argue that the odds of winning the lottery jackpot are 50:50 as you either win or you don't.
Just because there are only two possible outcomes, it does not follow that the probabilities are necessarily equal. As for what the whole Tuesday thing means I have no idea, but I know that my grasp of probability is so weak than any answer I come up with will be wrong. Probably.
Apologies to one and all for continuing it - I was merely trying to show how the precise wording of the question affects the answer.
Yeah I know - if you change the question you get a different answer. Stunning, is it not?
it's from this weeks new scientist
To answer the question you need to first look at all the equally likely combinations of two children it is possible to have: BG, GB, BB or GG. The question states that one child is a boy. So we can eliminate the GG, leaving us with just three options: BG, GB and BB. One out of these three scenarios is BB, so the probability of the two boys is 1/3.Now we can repeat this technique for the original question. Let's list the equally likely possibilities of children, together with the days of the week they are born in. Let's call a boy born on a Tuesday a BTu. Our possible situations are:
* When the first child is a BTu and the second is a girl born on any day of the week: there are seven different possibilities.
* When the first child is a girl born on any day of the week and the second is a BTu: again, there are seven different possibilities.
* When the first child is a BTu and the second is a boy born on any day of the week: again there are seven different possibilities.
* Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a BTu – and this is where it gets interesting. There are seven different possibilities here too, but one of them – when both boys are born on a Tuesday – has already been counted when we considered the first to be a BTu and the second on any day of the week. So, since we are counting equally likely possibilities, we can only find an extra six possibilities here.Summing up the totals, there are 7 + 7 + 7 + 6 = 27 different equally likely combinations of children with specified gender and birth day, and 13 of these combinations are two boys. So the answer is 13/27, which is very different from 1/3.
It seems remarkable that the probability of having two boys changes from 1/3 to 13/27 when the birth day of one boy is stated – yet it does, and it's quite a generous difference at that. In fact, if you repeat the question but specify a trait rarer than 1/7 (the chance of being born on a Tuesday), the closer the probability will approach 1/2.
Hmm seems to me that adding the Tuesday clause means that they end up counting BB twice because they now have extra information to differentiate between the boys.
Makes sense I guess.
Would you disagree?
no i was wrong 😳
on your original 33% 50% with one boy and then the first is a boy.
Your answers are correct. The gender of the next born child is unaffected by the gender of the previous child so it will always be 50/50.
(seems I dont do maths or logic) :
so if IGM can make that 50% argument for the eldest I think I can make it for the youngest, too
If so, it doesn't matter which the 1st mentioned kid is, the answer is 50%
So why isn't it 50% anyway ? 😳 😥
'Cos it isn't. It just isn't.
By the way if you like that one...
If a man walks up a hill, taking all day to do so, and camps at the top before walking down the next day, is there a point where he is in the same position as he was exactly 24 hours earlier?
You may all get the answer but it normally starts arguments.
If you care.
is the mountain on a conveyor belt? 😆
No but it comes with a crown race...
aaaaggghhhhh, FFS!
OK, how long is "all day"
how long does he camp for ?
How long does it take him to walk down ?
(trick questions?):
same route down ?
what do you mean by position ?
(seems I dont do maths or logic) :so if IGM can make that 50% argument for the eldest I think I can make it for the youngest, too
If so, it doesn't matter which the 1st mentioned kid is, the answer is 50%
So why isn't it 50% anyway ?
ok four outcomes M M , M F, F M , F F
with one being female we get M M , M F and F M so 1 in 3 of two males
If we know the eldest is male we have only M M and F M as options so 50/50 of two males
We reduce the number of combinations so change the odds.
24 hours assuming lots of things that just wont happen in the real world yes but not really as thye will be facing another direction unless they walk backwards
I'm sure there's some hormonal thing that ups the chances of the second being a girl if the first is.
Stat that 😛
well, yeah junkyard
but if you tell me that one of your kids is male and I say "If he's the eldest, there's a 50% chance you have 2 boys and if he's the youngest, thers a 50% chance you have 2 boys" how am I wrong ?
(I need something verbal to back up the maths because I'm just not feeling it for 1/3)
You're not wrong.
But if I don't tell you he's the oldest or youngest the odds are 33%.
Right about now you will be saying how can they be anything other than the oldest or youngest - but the knowledge about the family makes a difference to what you can say about them.
but if you tell me that one of your [two] kids is male
and I say "If he's the eldest, there's a 50%
chance you have 2 boys and if he's the
youngest, thers a 50% chance you have 2
boys" how am I wrong ?
You're not. You'd be correct.
Is the answer baby robin?
Sorry folks, I had planned on responding prior to this, but seeing as I got dragged away from t'internerd I couldn't get back on..
GrahamS, to answer your question, I am not sure as I wasn't the original one to ask the question, I heard it on Radio 4's More or Less, but it looks like it's made it into the New Scientist as well... it was originally asked at a maths/puzzle convference in the US.
However, I would guess that the Tuesday only relates to a the known boy.
Looking through the answer myself, I can see how the maths works out. Essentially the extra information you now have means that the order of a B B family is important with BTu B being different to B BTu unlike in the original where B B was just a single result.
Unlike Junkyard however, I think that this result is followed in the real world and would be backed up empirically. You should be able to reproduce this in a spreadsheet in just the same way as GrahamS did for the original.
As for the 50%ers... hehehe... now, I have these three doors, behind one of them is a stack of cash... ;o)
Unlike Junkyard however, I think that this result is followed in the real world and would be backed up empirically.
I dont agree but the reason the probabilities change is because we are only sampling a percentage of the TOTAL possible outcomes and the odss within that subset are as stated the actual odds have not altered.
MM MF FM FF
is reduced to MM MF FM hence the 1 in 3 as we ignore 25% of reality. I suspect the Tuesday does the same However I cannot really see how the dat affects anything in this example. In the first example the child must be born on a day being told which day it is has not altered anything in the real world. I agree them aths /spreadsheet will hold but it is not "real". It will not be observed by actually monitoring of reall births IMHO.
I agree them aths /spreadsheet will hold but it is not "real"
What if you used the spreadsheet to count a "real" collection of say 10,000 people that actually met the criteria? Would it become real then? 🙂
It will not be observed by actually monitoring of real births
As pointed out, real births are no where near this simple. The problem is set in a [i]statistically perfect world[/i], where every birth has a 50:50 chance of being either one boy or one girl and there are no troublesome factors like genetic predispositions, environmental influence, multiple births, IVF, XXY "girlboys", etc etc
At best you might see a rough approximation of this in the real world.
But yes, I'm reasonably convinced that if you monitored births within this imaginary construct then you would see the figures discussed.
What if you used the spreadsheet to count a "real" collection of say 10,000 people that actually met the criteria? Would it become real then?
That is my point it would be real but would come out as 33% not 13/27.
Yes my assumptions are made in a statistically perfect world as well.
Essentially you are arguing that the [b]day your child is born on affects the likelyhood of their gender[/b] and i say that is B0ll0cks for. The gender of the sibling alters the likelyhood of their gender I cannot see how day of birth does this as all children are born on a day] and i has has no bearing on likelyhood of gender.
I am astarting to get paranoid that I am doing a Smee here though 😉
I need to do asomemaths n this have we just reduced the set of all possibilities even further so that we have 13/27 but for less than 75% of the total births?
I understand your confusion Junkyard, and share it slightly.
I think the key is that [i]"The gender of the sibling alters the likelyhood of their gender"[/i] isn't really true. It just alters the likelihood that we are dealing with a child in a particular sub-group.
So the same applies to adding the day of the week information. It doesn't really mean that [i]"you are arguing that the day your child is born on affects the likelyhood of their gender "[/i] it just means the sub-groups we must consider have changed and so have the probabilities that you are dealing with a child in a given sub-group.
I think.
If I get time I might do a spreadsheet...
yes get the first bit we have removed one of the four posiibilitie shence 1 in 3 with you re sub set.
I agree with the next bit as well re the maths but I cannot believe thtat day of the week actually alters the probability of the gender after all in the first one the baby was born on a day of the week just because we now know which one I cannot see hwy th eprobabilities order.
I suspect the maths is correct but I dont agree with the conclusion reached that is that the odds have changed - your spreadsheet will show it as you describe no need to do this for me but if the inner Geek feels the need dont let me stop you 😆
My Inner Geek is strong:
http://spreadsheets.google.com/ccc?key=0Ai9dvO8nev8icF9INW8yU2VwM1BOeFBoQjFDamItRGc&hl=en_GB
The spreadsheet shows 1,000 pairs of siblings. It is 50:50 whether each is boy or girl and 1 in 7 for the day of the week they were born.
It should re-randomise them every time you refresh it.
I'm currently seeing: 119 pairs where at least one child was a Boy born on a Tuesday and of that 54 (45.9%) had a brother. 😀
Obviously it would be more accurate with a larger sample size, but I think it demonstrates the point 🙂
bows down at the alter of stats but you have linked to the old boy /girl one not the new one...unless i am as bad at computing as I am at getting this.
bows down at the alter of stats but you have linked to the old boy /girl one not the new one...unless i am as bad at computing as I am at getting this.
Try Sheet Two (tab at the bottom of the spreadsheet)
(did it work? did you see?)
sticks head in sand and refuses to play !
Yes and I dont fault the maths but I dont understand what has been removed to increase the odds. With the first example we loose FF hence the change what is it in the second that has been removed by knowing birth day? Will have a proper look / think tomorrow and you can help lead me to the light 8)
graham, can you easily add a cell or two to count how many (or, rather %) boys have:
older or younger brother
older or younger sister
❓
Yep, I'll do it tomorrow. But is should be around 50% for both.
Done, as expected:
Boys with brothers ~ 25%
Boys with sisters ~ 50%
All girls ~ 25%
Pretty much as per the original problem. 🙂
But is should be around 50% for both
😯as expected (25%, 50%)
Yeah, yeah, I misread your query yesterday. 😀
Anyway, does that help?
(no 🙁 ) cheers
Lol, oh well I tried. At least it demonstrates the maths is "real world"
scaredypants... try looking at it like this...
If you assume he is the eldest then there is a 50% chance the younger child is a boy
If you assume he is the youngest then there is a 50% chance the older child is a boy
That gives an overall 25% chance that the both children will be boys, which agrees with GrahamS's spreadsheets above, but this is the chance for 2 boys in the population as a whole.
By then using the knowledge that the family has to have at least one boy, that means that we can ignore all those families with 2 girls, which is 25%.
If we take the whole sample to be 100 families, this means that the number of families with 2 boys is 25/(100-25) = 1/3 or 33%.
Any better?
Funkynick - I'm way beyond help
I can "see" the statistical answer but I can't get it to agree with my own special logic*
*which would say that what you said about probabilty above should, if correct, also apply to him having an older or younger sister - which would leave us with a 33% chance of his sibling being neither a girl nor a boy 😐
Junkyard... you are not losing anything to increase the odds, nor is the day of the week having any effect on the likelihood of a child being born one sex or another... it's simply a matter of probability.
What happens is this...
In the previous problem, where we just know that one child is a boy, which gives the answer of 33%, we only know the following:
There are 2 children
One is a boy
The birth rate is an idealised 50:50 split
This means it only allows us to have four choices of how the family is made up:
GG BG GB BB
And so losing the GG option as we know there is at least one boy, gives us the 33% answer.
However, in the new problem, where we also know the boy was born on a Tuesday, this gives us extra information and so an extra option... so that now we have the following possibly options:
GG BTuG GBTu BTuB BBTu
Once again losing the GG option leaves us with the remaining 4, and the maths continues as posted on the first page, giving us the answer of 13/27.
So, by being given the additional information it means that the order of the birth for the two boys options becomes important, whereas previously, the order for the two boys was unimportant as all we knew was that there was at least one boy.
All this means is that as we get more and more information about the children involved, then the closer and closer the answer gets to being the 50% that a lot of people thought was the answer to the original question.
Any clearer?
scaredypants... thinking about it a little more, I have a feeling that this is one of those areas where I am not sure we can actually make those assumptions like that and then combine them unless both children are boys.
Stick with me here...
Due to the assumptions, firstly that there is an older boy in the first one, and that there is a younger boy in the second, these can only both be true if both children are boys, and that was how we got the 25% above, by multiplying the two 50% chances, so giving the chance that both those situations were true.
If we try it with older boy and younger girl and then younger boy and older girl, combining those would give trouble as both can't be true at once, so the chances of both happening at once is zero, so if one is true, then the other is not, and vice versa.
So, to get the answer for the probability of having a boy and a girl, you would subtract the probability of getting 2 boys from 100 to give the answer... 66%
I think...
This might be way out as I may be stretching my knowledge of probability just a tad here!