Brain Teasers and Maths Puzzles

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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Hmm... A math puzzle! I should be able to figure this out... :)

...

Well, this is either really easy or really complicated. My first thought is that the person who says yes either has 2 girls or a boy and a girl, so that would make the answer 50%, or half. If we're asking about the chances relative to the total number of people asked, then my answer gets more complicated. :P

Before I go any further, are we measuring the proportion of people who answer yes or of the total number of people asked?
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Re: Brain Teasers and Maths Puzzles

Post by Zyx »

Tormuse wrote:Hmm... A math puzzle! I should be able to figure this out... :)
So I thought too... having a degree in statistics and all that...
Well, this is either really easy or really complicated. My first thought is that the person who says yes either has 2 girls or a boy and a girl, so that would make the answer 50%, or half.
The answer isn't 50%, but it's not as low as eMTe's guesses either.
If we're asking about the chances relative to the total number of people asked, then my answer gets more complicated. :P Before I go any further, are we measuring the proportion of people who answer yes or of the total number of people asked?
No, either of those would be impossible to measure. You could calculate 95% confidence level of how much people I might need to ask before I found a number of people who would answer yes, but for that you would need to know what's the distribution of number of kids a random person has. I'm just asking for the probability of having two girls.
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Re: Brain Teasers and Maths Puzzles

Post by eMTe »

I was thinking along these lines: you need to know if somebody has two children, is one of them a girl and was she born on Sunday. I assumed that maximum number of children one can have is 2 (I don't know why I assumed this, because you didn't set a limit), so a person can have 0, 1 or 2 children - so there's 1/3 probability a person has two children. Now there's 1/7 chance that at least one of the children was born on Sunday. Also, if the person has two children - there's 1/3 possibility both are girls - because it can be either boy+boy, boy+girl or girl+girl.

1/3 * 1/3 * 1/7 = 1/63.

But I understand that the puzzle is tricky.
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Sorry, I phrased my question wrong earlier. I know we can't know the total number of people you ask, but generally speaking, in calculating probability, we need to compare the ratio of one particular event to all possible events. In order to answer the question of what the probability is, I need to know what events we're comparing here. Are we considering the ratio of families who say yes with two girls to the number of all possible combinations of genders and days? Or are we considering the ratio of families who say yes with two girls to the total number of families who say yes?

I'm still not sure I'm understanding, but after re-reading the question and looking at your hints, I'm going to guess that you mean the latter, and I now think the answer is 1/3 or 33.3%.

For each person who has two children, you have to consider the gender of each child. They can be boy+boy, boy+girl, girl+boy, or girl+girl. There are four possibilities and three of them have at least one girl. Of those three, only one has two girls. That makes the chance one in three.

Is that right? Or did I still misunderstand? :?
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Re: Brain Teasers and Maths Puzzles

Post by Zyx »

eMTe wrote:I was thinking along these lines: you need to know if somebody has two children, is one of them a girl and was she born on Sunday. I assumed that maximum number of children one can have is 2 (I don't know why I assumed this, because you didn't set a limit), so a person can have 0, 1 or 2 children - so there's 1/3 probability a person has two children.
The probability that the person has two children is 100%, because I already found that person.
Now there's 1/7 chance that at least one of the children was born on Sunday.
That's also 100%, because that person also answered that he or she indeed has a girl born on a Sunday. The only people who come into consideration are people with two children, with the added assumption that one of them is a girl born on a Sunday.
Also, if the person has two children - there's 1/3 possibility both are girls - because it can be either boy+boy, boy+girl or girl+girl.
Actually, 25%, because it's either boy+boy, boy+girl, girl+boy or girl+girl.
But I understand that the puzzle is tricky.
It actually isn't, but our intuition just sucks.
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Re: Brain Teasers and Maths Puzzles

Post by Zyx »

Tormuse wrote:Sorry, I phrased my question wrong earlier. I know we can't know the total number of people you ask, but generally speaking, in calculating probability, we need to compare the ratio of one particular event to all possible events.
Yes, and that's what we're after. What is the probability that given two children, other which is a girl born on a Sunday that the other child is a girl as well.
Are we considering the ratio of families who say yes with two girls to the number of all possible combinations of genders and days? Or are we considering the ratio of families who say yes with two girls to the total number of families who say yes?
Yes, we're looking after the latter, but with the added information that the other child is a girl born on a Sunday.
I'm still not sure I'm understanding, but after re-reading the question and looking at your hints, I'm going to guess that you mean the latter, and I now think the answer is 1/3 or 33.3%.

For each person who has two children, you have to consider the gender of each child. They can be boy+boy, boy+girl, girl+boy, or girl+girl. There are four possibilities and three of them have at least one girl. Of those three, only one has two girls. That makes the chance one in three.
This would indeed be the case, but you're forgetting that bit about Sunday.

Argh. Now I'm all worried that I worded the original question wrong, but I'm quite sure I have the information gathering in the right order. I really should stop posting puzzles that have longer Wikipedia pages than most western European countries.
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Aaaah! I just wrote out a reply, and then you posted again before I could submit it, so now I have to rethink what I wrote. :P
Zyx wrote: This would indeed be the case, but you're forgetting that bit about Sunday.
No, I deliberately omitted it. ;) If we're only focusing on the people who answered "yes," and excluding everyone else from our calculations, then the question of what day anyone was born is irrelevant, particularly because of this quote:
Zyx wrote:
Now there's 1/7 chance that at least one of the children was born on Sunday.
That's also 100%, because that person also answered that he or she indeed has a girl born on a Sunday
This seems to suggest that we are only considering the subset of people who said yes to having a family that includes a girl born on Sunday and ignoring everyone else, but...
Zyx wrote: This would indeed be the case, but you're forgetting that bit about Sunday.
this quote suggests that we are *not* ignoring everyone else. This is why I was asking clarifying questions about what we're including in our calculations, because it significantly changes how much work I have to put into my answer. :)
Zyx wrote: Argh. Now I'm all worried that I worded the original question wrong, but I'm quite sure I have the information gathering in the right order. I really should stop posting puzzles that have longer Wikipedia pages than most western European countries.
I'm guessing the Wikipedia page is so long because people are disputing the answer? If that's the case, then you probably got the wording right. It's just one of those questions that people have different opinions on, depending on how you approach it. If the answer is what I think it is, then I'm one of those people who would dispute it and I'll explain why later. I have a feeling I know what information you're looking for now, but this kind of calculation gives me a headache, so I'm going to come back to this later today, when I'm better rested. :D
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Re: Brain Teasers and Maths Puzzles

Post by eMTe »

Wouldn't this puzzle sound easier that way?

>>This weekend, I went out on the streets and asked random people two questions:

"Do you have two children?"

"Is one of them a girl born on Sunday?"

Finally I have found a person that had answered yes to both of my questions.

Everyone I asked had an equal chance of giving birth to either boy or a girl and I also know that births are equally distributed among the week. Also, the births are statistically independent so no twins or anything.

What is the probability that the person answering yes to both of my questions has two girls?<<
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Re: Brain Teasers and Maths Puzzles

Post by eMTe »

I read the puzzle once again and Ive found another source of potential misunderstanding. I don't understand what does mean that
Zyx wrote:births are equally distributed among the week
?

If it means that each week the same number of girls and boys is born that's fine, but this information is irrelevant when you want to calculate the possibility of having a girl as a second child for a single person - it still can be either boy or girl, so the chance is 1/2.
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

You forgot the editorial commentary of "It's a curse, really!" That was my favourite part! :D

As for the above comment about births equally distributed among the week, I read that to mean there are an equal number of kids born on Monday as on Tuesday, or on Wednesday, or on Thursday, etc.
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Okay, double post so people know I made another attempt at this. :P

As I established in my above post, for each family, we have to consider the gender of each of their two children, and we also (apparently) have to consider the day of the week each of the two children was born. So, each child can be one of two genders and be born on one of seven days; this makes 14 different possibilities for each child. In order to calculate the total number of possible combinations of genders and days for both children, we have to multiply those numbers together. 14 X 14 = 196 possible combinations. (Yikes!) :o

Now, of those 196 possibilities, 1/14 have the first child being a girl born on Sunday and 1/14 have the second child being a girl born on Sunday. Of course, those numbers overlap, and this is the part that gives me a headache, because I'm sure there's some mathematical formula that makes it easy to take those figures and calculate the total number of families with at least one girl born on Sunday, but I can't think of it right now, so... hmm... actually, it's easier than I thought; I just have to add them together. :P 14 + 14 = 28, but I have to subtract one to get 27, otherwise I would be counting the family with two girls born on Sunday twice. ;)

Similarly, for considering how many of those combinations have one girl born on a Sunday and the other a girl born on any day, there are 7 possibilities, so 7 + 7 = 14, but, again, I have to subtract one to get 13, or I'll count the two girls born on Sunday twice.

ANSWER:

So, that gets me my final answer: the probability of the family that says "yes" to having a girl born on Sunday having two girls is 13/27. About 48.1%. (Hey, my first answer of 50% was pretty close!) :D

Did I mention that this kind of puzzle gives me a headache? :Wallbang:

I sure hope I got it right this time. :) After Zyx confirms this, I'll share my other probability related puzzle that this one reminded me of, and then I'll explain why I would dispute the answer. ;)
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Re: Brain Teasers and Maths Puzzles

Post by Zyx »

eMTe wrote:Wouldn't this puzzle sound easier that way?
Sure, but where's the fun in that? :D
Tormuse wrote:So, that gets me my final answer: the probability of the family that says "yes" to having a girl born on Sunday having two girls is 13/27. About 48.1%. (Hey, my first answer of 50% was pretty close!) :D
And that's the correct answer.
I sure hope I got it right this time. :) After Zyx confirms this, I'll share my other probability related puzzle that this one reminded me of, and then I'll explain why I would dispute the answer. ;)
The original version of this puzzle has two boys and Tuesday, so I mixed it up to make it harder to Google. However, here's the Wikipedia page, and here are good summaries on BBC and ScienceBlogs.
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Re: Brain Teasers and Maths Puzzles

Post by eMTe »

It is categorised as paradox, so I sustain my opinion that it is tricky.
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Okay, so here's my puzzle, given to me by a math teacher back in high school about 15 years ago. (Yikes! That makes me feel old!) :o

Anyway, you have two coins. One of them is an ordinary coin with heads on one side and tails on the other. The other coin has heads on both sides.

You pick up one of the coins at random without paying attention to which one you picked up and toss it three times. It lands on heads each of the three times.

What is the probability that you picked up the two headed coin?
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Re: Brain Teasers and Maths Puzzles

Post by eMTe »

If it's another trick I assume it's not 1/2.
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Yeah, it's the same sort of trick as the previous puzzle, but it's simpler. At least there aren't 196 possibilities. :)
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Puzzles and Riddles

Post by fibonicci »

This is the most interesting puzzle i have ever played...loved it

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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

I'm not clicking that link until I'm sure it's not spam. :P
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Re: Brain Teasers and Maths Puzzles

Post by Tormuse »

Doesn't anyone want to try this? Do you need a hint? I promise it's simpler than the previous one! :D

Well, if no one else posts by the 16th, (1 week after I posted the puzzle) I'll post the answer and my comments about it. :)
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Re: Brain Teasers and Maths Puzzles

Post by Zyx »

I'm assuming this is a simple probability calculation using Bayes' rule, so that's why I don't want to answer.

A = The picked up coin has two heads (-A = A fair coin)
B = The coin lands heads up three times in a row
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A) = 0,5
P(-A) = 1-P(A)
P(B|A) = 1
P(B|-A) = 0,5^3 = 0,125
P(B) = P(B|A)*P(A) + P(B|-A)*P(-A) = 0,5625

So, P(A|B) = 0,89%
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