Critical thinking puzzle.

[Admin]

A couple have 2 children. One of the children is known to be a boy. What is the probability of his sibling being a girl?

Note: it's not a silly trick question.

[Lord Muck oGentry]

Which one is known to be a boy when both are boys { stirs pot maliciously >:D}?

[Admin]

The only info we have is that there are 2 children, one of which is a boy.

[Aardvark]

I striggle with these but here i go for broke.

The options are boy or girl. there is a fifty fift chance of any child being a boy or girl. The fact that they already have one boy has no bearing on the chance of the second child as they are unrelated events ( other than they are all related, ba dum tish.........I thankyew)

so my answer

50% chance of being a girl

Now someone smarter than me is going to shoot down my practical logic :(

[Blue Bubble]

Off the top-of-me-head: I'd say the probability that the other is a girl is 0.5 minus the probability of the occurrence of identical twins, plus the probability of the occurrence of twins, one of which is the opposite gender.

Wasn't there a long thread about this on the JREF last year (or thereabouts) ?

[Zendal Darkman]

My money is on "blue bubbles"answer.

however.....

For every 100 baby girls, there are 106 baby boys. This is a fact.

Why more boys? Nobody knows! And it's a real puzzle because...

Other animal species have half male and half female offspring.
Men produce equal numbers of x-chromosome sperms (which make girls) and y-chromosome sperms (which make boys).
from http://www.sexratio.com/facts.htm.

I'll allow others to check the sources of this 'amazing fact' ;D

[Admin]

The answer's not 50% ???

[Aardvark]

I striggle with these but here i go for broke.

The options are boy or girl. there is a fifty fift chance of any child being a boy or girl. The fact that they already have one boy has no bearing on the chance of the second child as they are unrelated events ( other than they are all related, ba dum tish.........I thankyew)

so my answer

50% chance of being a girl

Now someone smarter than me is going to shoot down my practical logic :(



Bugger, I have had a glass or two of the fermented grape and just realised that there are mosaics to consider

Turners Syndrome and Kleinfelters Syndrome are two that spring to mind

XO
XXY
XYY

Aslo from John's reply post that the answer is not 50%. is this in fact based on the demographics posted by blue bubbles??

[vbloke]

51% boy
49% girl
?

[Lord Muck oGentry]

The only info we have is that there are 2 children, one of which is a boy.


Fair enough, John. I take that to mean: it is known of one (unspecified) child that that child is male.
If it is a Monty Hall lookalike ( or Restricted Choice problem, for bridge players), then the answer is 2/3.

[Lord Muck oGentry]

Sorry. I've just looked at the question again. On the assumptions, the answer should, of course, be 1/3, not 2/3.

[Blue Bubble]

The only info we have is that there are 2 children, one of which is a boy.


Fair enough, John. I take that to mean: it is known of one (unspecified) child that that child is male.
If it is a Monty Hall lookalike ( or Restricted Choice problem, for bridge players), then the answer is 2/3.


Ah but it's not a Monty Hall lookalike. In the Monty Hall case, you have a priori knowledge that both the car and the goat(s) exist, and can make adjustment accordingly. In John's case, we have no a priori knowledge that a girl child exists.

[vbloke]

Doing a bit more thinking about the problem...

There are a number of possible answers to this.

Consider the collection of all couples with two children. Now, consider the following two experiments:

For each couple, pick one of the two children at random. If that child is a boy, what is the probability that the other child is a boy? (Answer: the probability that both children are boys is 1 out of 2.)
Note that to perform this experiment, we need to know in advance the sex of only one child.

For each couple, if either child is a boy, what is the probability that the other child is also a boy? (Answer: the probability that both children are boys is 1 out of 3.)
Note that to perform this experiment, we must know in advance the sex of both children.

Begin with a group of 100 families, each with two children, distributed as follows:

25 families with oldest child a boy and youngest child a boy.
25 families with oldest child a boy and youngest child a girl.
25 families with oldest child a girl and youngest child a boy.
25 families with oldest child a girl and youngest child a girl.

Of this group, there are 50 families in which the oldest child is a boy. Of those 50 families, there are 25 families in which the youngest child is also a boy. In other words, out of the group of families in which the oldest child is a boy, 50% have two boys.

From the same group, there are 75 families in which at least one child is a boy. Of those 75 families, there are 25 families in which the other child is also a boy. In other words, out of the group of families in which at least one child is a boy, only 33% have two boys.

This is a bugger of a problem as the wording is ambiguous and it can have several answers.

Someone check my reasoning, please!

[Admin]

They're great these puzzles aren't they? ;D

They key to solving it is recognising that we don't know whether the boy was born first or second; we simply know that out of the 2 children one of them is a boy.

There are 4 ways of having 2 children:

Boy - Boy
Boy - Girl
Girl - Boy
Girl - Girl

As one is a boy, the girl-girl option is ruled out which leaves:

Boy - Boy
Boy - Girl
Girl - Boy

So if one is a boy there's a 67% chance that the other sibling is a girl.

NOTE: If we know whether the boy was born first or second the chances are 50% as expected.

It's a good little puzzle as it shows how easy it is for intuition to be wrong.

[Lord Muck oGentry]

A lot depends on how we come to know that one child is a boy. Assuming we have asked to see a child at random, I counted the prior possibilities ( seen child first) :

1. B1 B2
2. B2 B1
3. B G
4. G B
5. G1 G2
6. G2 G1

The last three possibilities are ruled out when we see a boy. Of the remainder, twice as many give a boy pair as a mixed pair.

I have checked my answer with Lady Muck, and she is of the firm opinion that I can't count for toffee. And shouldn't be let out on my own... :) She makes it even money.

JJ's answer ( and vbloke's, if I'm following it correctly) is that the question is, in effect: what is the ratio of mixed pairs to boy pairs in the real world? ( 2:1)

So I have come to the conclusion that I don't understand the question :)

[Admin]

These type of questions tend to take on a complexity of their own. ???

It's also important to only deal with the information at hand.

It's interesting though, as if you're posing as a psychic and you know a person has 2 children, one of which is a boy, then stating that the other one is a girl will be correct 2 times out of three.

I wonder how many knowingly fraudulent psychics use such statistical tricks to tip the scales in their favour? :ponder:

[median]

Interesting stuff, John.
This does give an insight into how easy it is for people to arrive at the 'wrong' conclusion.
During the train of dialogue, the Monty Hall problem was mentioned.
This solution had all manner of academics and mathematics PhD's arguing against the correct answer.
A sobering thought then that even learned men need not have an intuitive grasp of such things. (Am I being skeptical enough?)
Practical example; Dr Roy Meadows and the cot death incidents.
His (mis) calculations were based upon calculating porobabilities for independent events!!

[Admin]

The thing I like about puzzles like this is that they show that intuition and common sense can actually be quite wrong.

Yes, Sir Roy Meadows made a major blunder in calculating the odds of multiple cot deaths in a family. He took the figure of 1 in 8500 as the odds of one cot death and then multiplied it by itself to work out the odds of two cot deaths in the same family - which he said were ~1 in 73,000,000.

His mistake was to ignore the fact that other factors could make it more likely to occur in the same family (genetics, housing conditions, etc.)

He treated each cot death as an independent event and that's why he came up with the wrong figure.

[fontwell]

Although you later clarified the question, I'd just like to say this. You originally said,

"A couple have 2 children. One of the children is known to be a boy. What is the probability of his sibling being a girl?"

Now the thing is that "One of the children is..." is ambiguous in everyday English and could be intended to mean "A particular child is..." or "A total of one child is ". These two meanings have different implications for the the total number of boys.

Now, when you follow that by saying "...the probability of his sibling..." you really are talking about particular child (i.e. 'him') and you are are asking the about the probability of that one child's sibling being a girl. When this sentence follows the ambiguous first sentence it implies that "One of the children" is in fact a particular one, because we subsequently talk about 'him'.

Therefore after extracting the everyday English language meaning from the original question, it is not really actually asking the question you intended.

I dare say a lawyer would try to have it that the question could mean what you wanted it to but one could equally well ague the other case, so at the very least it is unclear.

Interestingly, the difficulty people have with answering this sort of question in general is that they tend to re interpret questions to be the very one you have asked, so it behoves you to ask the question in an un ambiguous form such as

"A couple have 2 children. At least one of them is known to be a boy. A particular child is found to be a boy, what is the probability that he has a sister."

and give the punters a fighting chance, because its still easy to get it wrong.

[Admin]

Well it's always difficult to word questions properly as people do indeed interpret them rather than take in exactly what they say.

I think the key to this question is that the birth order is not revealed. If we know that the boy was first (or second) born then the chances of his sibling being a girl are 50% as expected by chance.

[doubting thomas]

I hate these sort of puzzles for two reasons,

1. i'm crap at solving them
2. its completely irrelevant to anything whatsoever, a bit like sudoku

[Admin]

Oh, I think these type of connundrums can be very educational. Once you get the answer (and understand it) it can show you a new way of grasping how things really work compared to how we think they do.

Shall we do the "all ravens are black" one? :D

:eek:

[vbloke]

Mary's father has five daughters Nana, Nene, Nini, Nono. What is the name of the fifth daughter?

[seren]

Mary :D

[seren]

Have you looked at that one about the plane? Not sure if it's critical thinking so much as knowledge of physics, but it was all over the net last year.


A plane is standing on runway that can move like a conveyer belt. The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in opposite direction).


Can the plane take off?