Sons or daughters?
While walking in a park you encounter an old friend from times gone by. He invites you immediately to dinner to do some catching up.
Between courses you need to use the bathroom, and you enter the hallway upstairs. The door you thought was the bathroom seems to be something else: Completely pink, dollhouses, toys etc. Some girls bedroom.
The next door you try was correct. On your return to the dinner, you notice an envelope lying on the doormat. You recognize it as a letter sent to parents of at least one boy, to advertise some summer camp.
Back at the table, your friend shares that he has in fact three children.
The question is: is it more likely that he has two boys and one girl, or two girls and one boy? Can you figure out the exact probabilities?
Between courses you need to use the bathroom, and you enter the hallway upstairs. The door you thought was the bathroom seems to be something else: Completely pink, dollhouses, toys etc. Some girls bedroom.
The next door you try was correct. On your return to the dinner, you notice an envelope lying on the doormat. You recognize it as a letter sent to parents of at least one boy, to advertise some summer camp.
Back at the table, your friend shares that he has in fact three children.
The question is: is it more likely that he has two boys and one girl, or two girls and one boy? Can you figure out the exact probabilities?
Labels: SharedPuzzle
posted by Psyflash at
11:20 PM
22 Comments:
I will go with 100% probability that he has 2 girls and one boy.
My reasoning is down to grammar...
He sees a bedroom - "Some girls bedroom." Plural - more than one girl. Pendants rule!!;-)
Should there be an apostrophe on the girls? If so, thats a whole different ballgame.
Then it's 50-50. We know he's got a boy and a girl, the last sprog is 50-50 girl or boy.
I bet it's not that simple.
Chris, I hope you are not going to bring in the fact that the UK birthrate for girls vs boys favours girls 51.2% vs 48.2% and it is not 50-50? :-0
I am not sure we have enough info to guess at anything other than 50-50.
I shall stick to my guns, although, I agree with you and suspect that the question isn't going to have such an obvious/easy answer...
Still, back to work, or at least what I try and palm off as work...
Hm.. as you guys can probably tell by now, English is not my first language :)
on the queston:
"Should there be an apostrophe on the girls? If so, thats a whole different ballgame."
Yes, it was meant to imply possession.
The answer is not 50/50. It was a ONE-girl-room and all of his children have a separate room.
As a hint:
Think not only how the story alters your chances, but maybe how the answer might have changed the probability of the story itself.
It is not a trick question. And has nothing to do with birthrates (in the sense that I assumed a 50/50 ratio).
Ok, here are my thoughts....
The probability of a letter being on the doormat relating to a male child at the time that you visited the house is much lower than the probability of opening a bedroom door relating to a female child.
This would appear to indicate that there are more likely 2 boys in the house, rather than 2 girls.
I can't see a way of calculating the probabilities precisely because there are too many unknown variables and assumptions, but my logic is thus:
I don't know much about Summer Camp letters, but I would guess that there is probably a window of around 6 weeks when they would be sent out.
This means that, assuming you happened to visit the house during that 6 week window, if there was one boy in the house then the probability of the letter being on the mat is 1 in 42 (2.4%), not allowing for other factors such as time of day when the post was delivered etc.
If there were 2 boys in the house the probability changes to 1 in 21 (4.8%) as two letters will arive in the 42 day period.
The probability of opening a door of a girl's bedroom is 1 in 3 (33%) if there is only 1 girl in the house and 2 in 3 (66%) if there are 2 girls.
So if there is 1 girl and 2 boys we have 2 events that have happened with probabilities of 33% and 4.8%.
If there are 2 girls and 1 boy then the 2 events have probabilities of 66% and 2.4%.
That's about as far as I can get with it at the moment, I expect someone on here can take it further.
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I say 2 girls, one boy...
only one letter to parents of boys
The following are the only possible ways of having three children in order (ie eldest to youngest) :
1) bbb
2) bbg
3) bgb
4) bgg
5) gbb
6) gbg
7) ggb
8) ggg
The only information we have is that he has at least one boy, at least one girl, and three children in total.
Combinations 2-7 above satisfy these conditions. Half of these consist of 2 boys and 1 girl and the other half are 2 girls and 1 boy.
As the question stands, we have no other information such as the relative ages of the children. With the question as written I would have to agree that the answer is 50%-50%. I look forward to being proved wrong!
I am quite impressed by the reasoning of DualAspect. You are absolutely right in the probabilities on the opening a door:
"
The probability of opening a door of a girl's bedroom is 1 in 3 (33%) if there is only 1 girl in the house and 2 in 3 (66%) if there are 2 girls.
"
You are however not completely correct in the reasoning on the letter. Only *one* letter would be sent, regardless of there being either one, or two boys.
therefore, you can completely factor out the prob of you seeing the letter. Since although there was not a 100% chance of you viewing it, this chance does not effect the probability of there being one or two boys. It *does* give you a 100% prob. that there is at least one!
@zaux:
assuming you reacted to reacted to DualAspect. Yes, you are correct :).
So ... do you guys feel I should have used better phrasing?
Hi Psyflash,
We don't have Summer Camps in the UK, but I assumed that maybe if there were 2 boys in the household, perhaps attending different schools, then there would be 2 separate letters sent to the parents. I guessed that this may be the clue to finding the correct answer as I can't see any other pointers in the text.
So the probability of the letter being on the mat has no significance to the solution of the puzzle, and in your 4:18 post you say that 50/50 is not the correct answer.
For now I'm completely baffled.
Is it simply that the fact that the opened door was a girls' room gives a greater probability of there being more girls than boys in the house?
On the opening of a random bedroom door, if there were 2 boys in the house the probability of it being a girl's room would be 33%, and 66% if 2 girls in the house.
The fact that the random door was a girls' room indicates that it is twice as likely that there are 2 girls in the house.
Therefore the probabilities of the distribution of boys v girls are:- 2 girls, 1 boy (66%), 1 girl 2 boys (33%)??
Well done DualAspect. That's the kind of thing I was thinking of when I responded to Karl.
Pedant strikes again:
"He invites you immediately to diner ..."
In my part of the USA, "diner" is a type of restaurant, not one of the daily meals. Therefore there should be NO bedrooms and NO personal mail in sight. :-)
Thanks Chris.
Once you see the light this is an obvious variation on the Monty Hall problem, but the eroneous irrelevant info threw me for a while.
I bet you got it straight away and sat quietly chuckling to yourself reading the rambling guesses above?
DualAspect. I didn't see the problem until 8 am, I decided to go to bed instead of having a go. The two pieces of evidence were too much much for my shutting down brain.
Psyflash. The only unfortunate bit was the inclusion of "some" and the omission of the '. I would have phrased it as "a girl's bedroom".
Karl. I wouldn't unnecessarily introduce UK demographic data on an international website. The basic logic is hard enough, introducing such refinements simply clouds the issue.
There are many possible solutions to this question.
1. It is 50% for both, because 1 girl and 1 boy are there therefore a 50% for the other to be either.
2. Your friend is a liar
3. There is a 0% for 2 girls and 50% for 2 boys probability, because the people who owned the house before could have had a girl and your friend might have just moved in. Therefore there could be 2 boys. The boy that lives in the girlish room, the boy who got the letter, and the other child that could be either boy or a girl.
4.The letter could have also been sent to the wrong house on accident, because the mail carrier made a mistake and sent it to the wrong house. Therefore there could be 3 girls or 2 girls and 1 boy.
5. A strange combination of answers 3 and 4 could make it swing either way due to probability and because the probabilities are unknown makes the answer unpredictable.
6. Your friend is actually your wife who is also your daughter, because you used to be married to her mom. You had your future wife as a child. Devorced your future wife's mom and married your daughter. Therefore she counts as a child in the family. Therefore when she tells you she has 3 children you actually have 4 creating a paradox.
There are many more possible scenarios, but these ones seemed the most suitable.
Posted by The Godson
You said he went to the bathroom.. Was the seat up, or down?
DualAspect:
"
Therefore the probabilities of the distribution of boys v girls are:- 2 girls, 1 boy (66%), 1 girl 2 boys (33%)??
"
Yes this is correct. I feel that you and Chris both have the underlying logic pinned down.
I did not say that the letter was irrelevant to the story, just that the event of you seeing it has no influence on the prob of the last boy/girl, It only assures you of the fact that he has at least one boy!
The reasoning you posted, on the chances of accidentally stumbling upon the room is the relevant part of the puzzle.
The Thomas referred to in the title is of course Thomas Bayes.
Euclid's Brother: Why do you care if the seats up or down? No offense.
If the seat is up, then it's increases the probability that there are more males than females in the house.
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why do only boys go to summer camp??????
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