Probability Puzzle

[K^2]

On 7/18/2018 at 2:20 AM, Gargamel said:

This is incorrect, as the wording in the puzzle is key.  It's not "what are the odds the other child is a boy", but "is also a boy".

Wording doesn't make any difference here. The only question is how, if at all, you discard potential matches from the random sample. This is very easy to confirm with simulation as well.

If you simply walk by groups of parents with two children, discarding any where at least one of the children isn't a boy, yes. You will arrive at the answer you're hinting at, because you've actually completely excluded 1/4 of groups from your sample. But that isn't inherent in the statement of the problem. You could interpret it as that you've ran into the first group with two children, picked one of them, and said, "one of the children is a <gender>, what are the odds that so is the other?" And here, we are back to 50/50, because you did not discard anyone.

The question is entirely ambiguous on this matter, which is why I absolutely hate this puzzle coming up. It tends to start a crap fight between two groups of people with rather rudimentary understanding of statistics, nor willing/able to write a simulation to back up what they are claiming.

[wumpus]

4 hours ago, mikegarrison said:

These two parts of your answer illustrate the same thing: word problems are stupid. Textbooks start with a mathematical equation and then try to build a "word problem" around it. It's almost never anything that makes real world sense, which everybody recognizes and many people joke about.

In real life, many engineering problems are "word problems" (example: "how thick does the skin of this airplane need to be at this location of the fuselage?"), but none of them look or sound like a math textbook word problem. Instead we get "Amy has six apples and five oranges. If she can trade two apples for one tomato and five oranges for two tomatoes, how many tomatoes does she have after the trades?" Word problems are stupid because they aren't teaching you how to apply math to the real world -- they actually try to apply the real world to math.

You've nailed it.  Unfortunately, word problems have sounded the same since day 1 (I forgot whether it was the Islamic-era work that the name "Algebra" comes from or an earlier mesopotamian work, but the algebra problems in translation sounded exactly like modern "word problems").

I'm not volunteering to fix it.  Trying to come up with problems that are appropriate for the class to solve while ignoring the obvious math underneath seems exactly like the classic "don't think of polar bears" problem.

The classic counterexample is "what's 2*7+3?".  "Don't know."[US specific content follows] "what's two touchdowns and a field goal?" "17."[/US content].

[sh1pman]

So, what’s the right answer then? 1/3, according to Bayesian math tricks, or 1/2 or so, according to common sense?

[mikegarrison]

16 minutes ago, sh1pman said:

So, what’s the right answer then? 1/3, according to Bayesian math tricks, or 1/2 or so, according to common sense?

It depends on how you read the question. I suspect the intended right answer was 1/3.

[Cunjo Carl]

8 hours ago, mikegarrison said:

 Word problems are stupid because they aren't teaching you how to apply math to the real world -- they actually try to apply the real world to math. 

Hm.  You actually really nicely explained something that's always bothered me about modern text books as an instructor that I could never quite put into words. I think in this particular case, the question was presented as a puzzle, so I wasn't surprised when it tricked me!

 

ED: Wait. I have it! Finally got to the bottom of it. 0%! Because we already established that she has two children and (precisely) " One of them is a boy." . Since only one is a boy, of course the other can't be! 

 

Edited July 20, 2018 by Cunjo Carl

[Xd the great]

50%. CASE CLOSED.

[monophonic]

D'oh. Obviously the other kid exists in a quantum superposition of a boy and a girl. Therefore the probability of him being also a boy is 1 until we try to observe him. At that point he will collapse into one or the other gender and is not also a boy any more.

 

 

[mikegarrison]

Don't forget that roughly 1 in 1500 people are born intersex. Gotta include that in the probability tables.

[YNM]

11 hours ago, K^2 said:

"one of the children is a <gender>, what are the odds that so is the other?"

What, I'm not allowed to filter the database by changing the "<gender>" to "definitive boy (XY)" ?

 

And on "oh, it's a word problem again" :

How in the hell do you think the world works ? Miming ?

These are real world problems where language was a "culprit".

Now, it is possible to make a pictoram (or even 'silent' comic) for the question, but that's cumbersome.

And we haven't even talked how much more ambiguous emotion symbols are (which is why emojli [this] didn't work).

Edited July 20, 2018 by YNM

[K^2]

8 minutes ago, YNM said:

What, I'm not allowed to filter the database ?

You are. We're trying to decide between "SELECT * FROM encounters WHERE child1 = 'boy' AND child2 = 'boy';" and "SELECT * FROM encouners WHERE child1 = child2;"

[YNM]

4 minutes ago, K^2 said:

We're trying to decide between "SELECT * FROM encounters WHERE child1 = 'boy' AND child2 = 'boy';" and "SELECT * FROM encouners WHERE child1 = child2;"

It's definitely the first.

But after an "at least one is boy".

[K^2]

15 hours ago, YNM said:

It's definitely the first.

I don't disagree that it's AN interpretation. I just don't see it as the only valid way of reading the problem statement. The "At least one is a boy" could have been a pre-selection criterium, or it could be reaction, with "At least one is a girl" having been just as likely. We can't tell which it is from the problem statement, but it's a critical piece of information to gauge the probability.

[Xeorm]

1/2, assuming we're not worrying about minutiae like men being born more often then women. Unless it's something like this:

https://xkcd.com/169

[YNM]

1 hour ago, K^2 said:

The "At least one is a boy" could have been a pre-selection criterium, or it could be reaction, with "At least one is a girl" having been just as likely.

I guess it's slightly Monthy Hall -esque.

I like to think the problem presented here as being a two-filter case. (so yeah, 1/4 of the 'data' is discarded.)

[Cheif Operations Director]

On 7/18/2018 at 4:01 AM, Gargamel said:

Posting this in the S&S forum, rather than the Lounge, so we can have a intelligent conversation about probabilities vs Bayesian Probabilities.   If you know the answer, or find this trivial, just don't blurt it out, let the others figure this out.    When we've had a bit of good discussion here, I'll post up the second puzzle. 

 

So here's the Question:

 

You are walking down the street, when you run into a friend of yours.  She has her two children with her.  One of them is a boy.  What are the odds that the other child is also a boy?

50/50 If "other refers to the child that is not boy 1. 

[Cheif Operations Director]

 

If child 1 is a boy child can has only be a boy or girl hence 50-50 propobility. You can not take chance that it may be 51% boy 49% girl as mentioned by @5thHorseman that is a chance statistic not a fixed number or ratio. Use of the word "also" changes nothing either 2 children each with a 50% probability because of the two options. This is differant than the game show example too that has a 1/3 probability this only has 2 options and hence 50% probability. 

 

[Xd the great]

As we know that child 1 is a boy, chances of both being boys are 50%

[K^2]

8 hours ago, YNM said:

I like to think the problem presented here as being a two-filter case. (so yeah, 1/4 of the 'data' is discarded.)

I just don't see it being implied in the wording. Imagine that the teller has decided in advance that he'll pick the gender for the intro based on the first child they see. It happened to be the boy, so it was "one of the children was a boy, what are the odds that so was the other." If that first child happened to be a girl, the teller would instead ask "one of the children was a girl, what are the odds that so was the other". It just worked out that it was a boy in this case.

The statement of the problem doesn't tell us if the teller decided in advance that they'll be only considering groups where at least one child is a boy, or if they chose to say "boy" or "girl" based on observation. And that changes the outcome, because now half of the encountered groups will have resulted in "at least one is a girl" telling, and since we have the "at least one is a boy" telling, they are discarded. So we're left with 1/2 of initial sample instead of 3/4, and within that remaining 1/2, the odds of other child being boy or girl are 50/50.

Once we decide on interpretation, this is a very simple problem. But choice of interpretation is ambiguous with respect to wording. There is nothing in the problem telling us that choice to ONLY consider groups with at least one boy was made apriori.

[YNM]

12 minutes ago, K^2 said:

Imagine that the teller has decided in advance that he'll pick the gender for the intro based on the first child they see.

Unless this is not the " 'first' child". It's just a possibility of anything that there's a boy. I don't care which one.

 

In fact, the same would've worked if the wording was to be changed "one is a girl, what is the chance the other is also a girl ?".

[magnemoe]

On 7/19/2018 at 10:59 PM, wumpus said:

You've nailed it.  Unfortunately, word problems have sounded the same since day 1 (I forgot whether it was the Islamic-era work that the name "Algebra" comes from or an earlier mesopotamian work, but the algebra problems in translation sounded exactly like modern "word problems").

I'm not volunteering to fix it.  Trying to come up with problems that are appropriate for the class to solve while ignoring the obvious math underneath seems exactly like the classic "don't think of polar bears" problem.

The classic counterexample is "what's 2*7+3?".  "Don't know."[US specific content follows] "what's two touchdowns and a field goal?" "17."[/US content].

2*7+3=14+3=17, the rules is that you multiply before adding. Math has precise rules we have defined, 1/0=infinite is one, I hate it I rather want 1/0=0 as that would not crash computer programs if you ask for average price of cars sold a day you did not sell any. 
Language is not precise and its very easy to make it less precise just listen to politicians and salesmen, legal texts try to make it more precise but also face definitions as in that is legal self defense and reckless driving. 

[Cheif Operations Director]

It's 50%! 

 

[Chakat Firepaw]

1 hour ago, magnemoe said:

Math has precise rules we have defined, 1/0=infinite is one, I hate it

You are right to hate that, because it's wrong.

1/0 isn't infinite, it's undefined.  Outside of some very special cases, zero is non-negative but not positive.

[YNM]

2 hours ago, magnemoe said:

... Math has precise rules we have defined...

 

[Cheif Operations Director]

25 minutes ago, Chakat Firepaw said:

You are right to hate that, because it's wrong.

1/0 isn't infinite, it's undefined.  Outside of some very special cases, zero is non-negative but not positive.

It's neutral

It's 50% anything else is illogical. Grammar does not apply here and we have only two options. It's like saying a house is over that  hill (boy) or it is not (girl) it 50% you may have a house or you may not. Birth rates are not statistly valid, they change every second, literally. With no other useful information what are we to deduct other than a 50% - 50% probability.

[K^2]

16 hours ago, YNM said:

Unless this is not the " 'first' child". It's just a possibility of anything that there's a boy. I don't care which one.

 

In fact, the same would've worked if the wording was to be changed "one is a girl, what is the chance the other is also a girl ?".

Yeah, but if you choose to say "one is a boy" or "one is a girl" after you saw the group, based on whether there is at least one of these, then we're back to 50/50. That's crucial.

Edit: Specifically, consider this setup. I walk on the street. I see a group with two children. I pick one child at random, if it's a boy, I say, "At least one is a boy." If it's a girl, I say "At least one is a girl". The choice is entirely random, and so it's not "first" or "older child" or any of that nonsense.

Do you agree that this is consistent with wording of the OP's problem? If not, then could you please clarify what the difference is? Could be purely linguistic disagreement. But if you agree that this setup is consistent with wording of the problem, it's trivial to show that in THIS setup, the other child can be a boy or a girl with 50/50 odds.

Edited July 22, 2018 by K^2