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Think you understand Monty Hall? Try the Tuesday boy problem.

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Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#121

Post by ghshephard »

"You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.” What is the probability that the other child is also a son?"

The fact that one of the sons is born on a Tuesday, is blonde, has freckles, got an A+ on his math paper is irrelevant to the gender of the second son. The author's entire reasoning misses the entire point of the Monty Haul paradox where Monty Haul specifically reveals a door known _not_ to have the Goat, thereby immediately modifying the probabilities of the remaining door.

In this "Tuesday Boy Problem" - no such selection takes place. There is no paradox. There is a 50% chance that the other child is a boy and a 1/7 chance that they were born on a tuesday (or a monday, wednesday, etc...).

Note - The wikipedia article on this topic does a much better job discussing the ambiguity involved in asking the question - http://en.wikipedia.org/wiki/Boy_or_Girl_paradox.

The sad part of this, is that if you had _selected_ a family in which at least one son was born on a tuesday, then you would have modified the probabilities of the other child being born a son - but no such selection was done here, therefore no probabilistic impact on the chance of the other child being a son.

Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#122

Post by monorailkitty »

reitzensteinm wrote: post #58

I've been thinking about this for an hour, and I'm now convinced that the author is wrong. The fact that we found out about one of the children from the father means that all probabilities are not equal, even though they're treated here like they are. The difference is between the information being offered, and determined independantly. I'll do this with the boy/girl problem, for simplicities sake. If we ask a man if…

Yeah, what matters is the contents of the initial set of families over which we determine probability.

"A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?"

If the man is randomly chosen from the set of all families the answer is 1/2.

If the man is randomly chosen from the set of all families with a son born on a Tuesday then the answer is 13/27.

The reason for the difference is that a boy/girl family has a 1/7 chance that the boy was born on a Tuesday whereas the boy/boy family has only a 13/49 chance.

B G (7/49 probability of a Tuesday boy)

G B (7/49 probability of a Tuesday boy)

B B (13/49 probability of a Tuesday boy)

13 / (7 + 7 + 13) = 13/27

Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#123

Post by ajanuary »

ars wrote: post #65
ajanuary wrote: post #62

Earlier quoted context omitted.

Looking at your simple example of genders using the same process the article uses to enumerate the combination of gender and days: Assuming Child 1 is a Boy: Child 2 can be: Boy, Girl Assuming Child 1 is a Girl: Child 2 can be: Boy, Girl Assuming Child 2 is a Boy: Child 1 can be: Boy, Girl Assuming Child 2 is a Girl: Child 1 can be: Boy, Girl Combining those gives us the following combinations: Child 1 | Child 2 ----…

Except you are not actually supposed to remove the duplicates!! It is a very common mistake, but it's simply incorrect, it leads to incorrect results. Also, why are you numbering the kids as child 1/2? There is no such distinction made. If you changed your list so that the fixed child is always listed first, and removed duplicates you would have 4 possibilities.

I would be interested in your reasoning why G,B and B,G can (rightly) be considered distinct, and B,B and B,B can (rightly) be considered the same enumeration and thus one discounted, but BT,BT and BT,BT () should still be considered distinct when it is clearly the same situation as B,B?

() Where BT is "Boy born on a Tuesday"

Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#124

Post by danparsonson »

ajanuary wrote: post #117
danparsonson wrote: post #88

Earlier quoted context omitted.

I think I've spotted where your misunderstanding is. BG is only not the same as GB if there is some other information available - which was born first, what their names are, hair colour, etc., because then you'd be saying something like Boy born first, Girl born second Girl born first, Boy born second and those are two distinct possibilities. The point is that they are only distinct if you have this extra information…

No, they are correct in saying that GB and BG are distinct, even with no other information. It is not order that is important, but considering each child as a distinct entity. The 50% chance of being a boy and 50% chance of being a girl applies to a single independent child. When enumerating the possible combinations we need to first enumerate the possibilities for each child, and then combine these two enumerations…

Hmmm I think we're saying more or less the same thing. The point I wanted to make is that

"I have a son and a daughter"

is the same as

"I have a daughter and a son"

unless you qualify the statement with further information - names (Alex and Sam from your post) would be an example of that further information. My feeling was that ars is implicitly 'filled in the blanks' somewhere, treating the two children as distinct when in fact they have to be interchangeable for the purposes of the original (13/27) calculation.

It's the lack (or not) of such details that alters how the probability is calculated.

I admit though that I'm still chasing myself in circles trying to understand the whole thing so take this reply with a pinch of salt ;-)

Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#125

Post by _dps »

ghshephard wrote: post #121

"You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.” What is the probability that the other child is also a son?" The fact that one of the sons is born on a Tuesday, is blonde, has freckles, got an A+ on his math paper is irrelevant to the gender of the second son. The author's entire reasoning misses the entire point of the Monty Haul paradox where Monty Haul specifica…

There is a shorthand and implied understanding in mathematical word problems (else they would be formulas!). As soon as you ask "what is the probability" you are implying either a sampling/generative process, or a subjective (e.g. Bayesian) probability framework. I think it's clear this is the former case.

The generative model implied for each child is: pick uniformly from boy/girl, and pick uniformly from Mon-Sun. The generative model for the father is: generate two children.

This generative process has a well-defined outcome distribution and it is completely reasonable to ask "what is the probability of generating an outcome with two boys, given that you generated an outcome with a Tuesday boy?"

Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#127

Post by anorwell »

monorailkitty wrote: post #122
reitzensteinm wrote: post #58

I've been thinking about this for an hour, and I'm now convinced that the author is wrong. The fact that we found out about one of the children from the father means that all probabilities are not equal, even though they're treated here like they are. The difference is between the information being offered, and determined independantly. I'll do this with the boy/girl problem, for simplicities sake. If we ask a man if…

Yeah, what matters is the contents of the initial set of families over which we determine probability. "A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?" If the man is randomly chosen from the set of all families the answer is 1/2. If the man is randomly chosen from the set of all families with a son born on a Tuesday then the answer is 13/27. The…

>If the man is randomly chosen from the set of all families the answer is 1/2.

By assumption, the man has a son born on Tuesday, so this is hardly relevant.

If A is a subset of B, then choosing x uniformly at random from A given that x is in B is the same as choosing uniformly at random from B.

Re: Think you understand Monty Hall? Try the Tuesday boy problem.

#130

Post by jerf »

NickPollard wrote: post #87
reitzensteinm wrote: post #58

I've been thinking about this for an hour, and I'm now convinced that the author is wrong. The fact that we found out about one of the children from the father means that all probabilities are not equal, even though they're treated here like they are. The difference is between the information being offered, and determined independantly. I'll do this with the boy/girl problem, for simplicities sake. If we ask a man if…

The issue here is with assumptions - you have made a different set of assumptions from the author, and hence are getting a different result. A lot of people here are having similar issues, by misreading exactly what the initial proposition means. Your reasoning above relies on the 'likeliness' of a man giving you the information, which is something that is not meant to be a part of the problem. Although it is phrased…

I'd also observe that carefully read, this article is really about how important assumptions are, and not about the problem per se. The Peter Winkler quote is key.
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