Probability Fun

[omk3]

Thank you thank you. I'd have never gotten that far without all the helpful hints though.
My first thought looking at the Tuesday thing was "Huh?"

[LazyScot]

Originally Posted by LazyScot

Actually, I think this moving us on.

The first answer of 1/4, I'm confident of, since that is based on the importance of order (something tells me this is related to the difference between permutations and combinations....)

Anyhow, my first attempt at the second answer ignored order, and I think that was wrong, whereas yours did take it into account. Thus I think that the second answer should be 1/3 as the possibilities are BB BG GB.

If we were to expand the possibilities up each pair would come out to 49 possibilities. B(monday)B(monday), B(monday)B(tuesday), etc. However, we know that at least one boy was born on a tuesday, so the BG can only be expanded up to 7 possibilities each. B(tuesday)G(monday) and so on. By comparison, the BB expands up to 14 possibilities. So we have a total of 28 possibilities.

So the answer to 3 is, I think 1/2. (i.e. 14 out of 28)

I counted the B(tuesday)B(tuesday) twice. So the BB should expand to 13 possibilities. Hence a total of 13 out 27.

Originally Posted by omk3

Ok next try.
We have three possible combinations, GB, BG, BB.
The chances of the boy being born on a Tuesday on each of the first two cases is 1/7.
In the BB case, we have 49 possible combinations, in 13 of which there is at least on Tuesday. So the chances of at least on boy being born on a Tuesday in a BB case are 13/49.
Which makes the overall probability....

...13/27?

Oh how I wish I remembered how to calculate these things...

Originally Posted by pdurrant

Yes! We have a winner! 13/27 is indeed the right answer.

So can you confirm if my logic is right, or have I committed the ultimate sin (according to my old Maths teacher) of getting the right answer for the wrong reason...

[pdurrant]

Now, the article I mentioned gives the same answer but asks the question in a different way. And I think that with the question they asked, you don't get this answer. I'd be interested to hear other people's views on it.

Their set up was:

Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

In my opinion, the probability here is still 1 in 3. My argument is that the information that one of his children is a boy born on a Tuesday doesn't actually give us any new information. The Tuesday is arbitrary, and could equally well have been any day of the week.

If we had a large number of fathers of two children one of which is a boy, all of them could all say similar statements (with different days of the week). This can't mean that 13/27ths of them had two boys.

Of course, 27 in 147 could say exactly what Gary said, and if we examined those 27 in 147, 13/27ths of them would have two boys.

But when volunteering a day of birth, only 21 in 147 would pick Tuesday (assuming that births are equally distributed among the days of the week, and that people with two boys don't have a preference for giving one day of the week over another). And of those 21 in 147, only 1/3rd would have two boys.

It is for this reason that I argue that the New Scientist article got it wrong - volunteering the day of birth doesn't get us a subset of the population with an 'excess' of two boys. Only by question and answer do we remove from consideration more single boy families than two boy families.

[pdurrant]

Originally Posted by LazyScot

I counted the B(tuesday)B(tuesday) twice. So the BB should expand to 13 possibilities. Hence a total of 13 out 27.

So can you confirm if my logic is right, or have I committed the ultimate sin (according to my old Maths teacher) of getting the right answer for the wrong reason...

Yes, you had the reasoning right. It was just the counting of TBTB twice that fooled you.

Hopefully the chart will make things clearer.

[omk3]

Originally Posted by LazyScot

So can you confirm if my logic is right, or have I committed the ultimate sin (according to my old Maths teacher) of getting the right answer for the wrong reason...

Sounds like you got it first
And your reasoning went completely over my head yesterday when I read it. Go figure

[pdurrant]

A secondary question:

What would the probability of Dan having two boys be if he had answered "No"?

[obs20]

Originally Posted by pdurrant

Dan: I have at least one boy

Wouldn't this mean that from this point forward what are the odds of me having two boys, irregardless of birth order? Which would be 1/2

[pdurrant]

Originally Posted by obs20

Wouldn't this mean that from this point forward what are the odds of me having two boys, irregardless of birth order? Which would be 1/2

Umm.. no. If Dan had said "My first child was a boy" (or, of course, "my second child was a boy") then the probability of him having two boys would be 1/2.

But since Dan does not specify which of his children is the boy he's mentioned, the odds of him having two boys is 1/3.

With no other information than that Dan has two children, we know that his children are one of four possibilities, giving eldest first: Boy-Boy, Boy-Girl, Girl-Boy, Girl-Girl. We /are/ making the assumption that each of these possibilities is equally likely - i.e. that the probability of any particular child being a boy is exactly 1 in 2 and that birth order doesn't affect this probability.

When Dan says "I have a least one boy" this is equivalent to saying "I don't have two girls". It only eliminates one of the four possibilities, leaving use with three, equally likely, remaining possibilities.

If Dan said "The eldest is a boy", that would eliminate two of the possibilities (GB GG) leaving us with only two remaining, equally likely, possibilities.


[EDIT: See my later post where I explain why we're both wrong - I wave the wrong answer, and you have the right answer for the wrong reason ]

[obs20]

Originally Posted by pdurrant

Umm.. no. If Dan had said "My first child was a boy" (or, of course, "my second child was a boy") then the probability of him having two boys would be 1/2.

That's where I got confused. Looking at my notes I see that I did have 13/27 originally but then I over thought the problem. Good puzzle!

[dreams]

Congratulations, omk3!

Thank you for the chart, pdurrant. It helps greatly in understanding. (I just couldn't go to work without checking in for the answer. )

[Bilbo1967]

Thanks pdurrant. I couldn't have been more wrong if I tried but enjoyed thinking about it anyway.

The chart and other people's explanations helped let me see where I went wrong.

[pdurrant]

How embarrassing. I've just sneakily edited the question in the first message in this thread to fix the setup so that the answer I've given is right.

I was reading the comments over at New Scientist and realised that for much the same reason I disagreed with their answer to their setup for the final stage, I must also disagree with the other answer.

My original post had Dan stating "I have two children" and "I have at least one boy".

The first statement is OK, but for exact clarity hould have been (and now is)
"I have exactly two children".

The second statement doesn't work the way I wanted. I should have (and now have) turned it into a question by Nick and an answer of "Yes" from Dan. Here's why.

Consider a large set of fathers of exactly two children. They are all asked to make a statement of the form "I have at least one [boy | girl]". They can all do so. This doesn't mean that of the half that answered "I have at least one boy", only one third have two boys. Obviously on average half of that half have two boys, because 1/4 of the whole set has two boys.

In short, the answer at that stage is indeed 1/2 (as some said), not 1/3. Only if someone else asks the question and gets an affirmative answer does the probability become 1/3.

How very complicated. And how embarrassing that having spotted the problem with the second half of the New Scientist setup I failed completely to spot the problem with the first half.

[omk3]

Well, I may have found the answer but probabilities still confuse me. 1/2 seems to me as valid as 1/3, and alternative solutions proposed for the third stage sounded equally valid from a specific viewpoint.


I don't really see why the exact same information would produce different probability results depending on how you got the information. The hard facts we get from the Scientific American puzzle and pdurrant's puzzle are exactly the same. Does it really matter how they are presented?

[LazyScot]

Originally Posted by omk3

Well, I may have found the answer but probabilities still confuse me. 1/2 seems to me as valid as 1/3, and alternative solutions proposed for the third stage sounded equally valid from a specific viewpoint.


I don't really see why the exact same information would produce different probability results depending on how you got the information. The hard facts we get from the Scientific American puzzle and pdurrant's puzzle are exactly the same. Does it really matter how they are presented?

One way of looking at it is that by knowing at least one child is a boy, the probabilities of having a girl child *for that specific person* are no longer 1/2. If they were, they would still have a chance of having 2 girls. What has happened is that we have information that places the person in a subset of the overall population, so the probabilities change.

I think!

[pdurrant]

Originally Posted by omk3

I don't really see why the exact same information would produce different probability results depending on how you got the information. The hard facts we get from the Scientific American puzzle and pdurrant's puzzle are exactly the same. Does it really matter how they are presented?

This is a very good summary of the meta-problem. In both cases, we end up knowing that a particular person has two children, and one of the is a boy born on a Tuesday. It does seem peculiar for me to say that in one case the probability that the other child is a boy is 1/2 and in the other it's 13/27ths.

Let me give some concrete numbers to show what's going on. I'm going to use low numbers, and expect the probabilities to work exactly. Of course, in an actual trial it wouldn't work with such low numbers. Just imaging that when I say "196" I mean "196 million", etc.

We take 196 fathers of exactly two children.
I think we can agree that we'd expect 49 to have BB, 49 to have BG, 49 to have GB and 49 to have GG.

We ask them all to make a true statement about their children using the template "I have at least one [boy|girl]"

On average, we'd expect 98 of them to state "I have at least one boy" (because for 49 (BB) this is the only true statement they can make, 49 (GG) can't state this, and of the remaining 98 (BG, GB) we'd expect half to choose the boy statement and half to choose the girl statement)

Now we ask those 98 (49 BB and 49 (BG or GB) ) to make a true statement about their children of the form "I have at least one boy born on a [day of week]"

On average, we'd expect 14 to say "I have at least one boy born on a Tuesday". ( and 14 to say "..Wednesday, 14 say "Thursday", etc) And we'd of that 14, we'd expect to have 7 BB and 7 (GB or BG).

So now we have 14 fathers who've all made a true statement about their children "I have at least one boy born on a Tuesday". How many of them have BB? Exactly 1/2.

Now, let's take that same 196 fathers we started with. This time we /ask/ them "who has at least one boy?". This time we expect to get 147 saying yes - all 49 BB, 49 BG and 49 GB.

And now we ask those 147 "who has at least one boy born on a tuesday?"
We expect to get 7 BG and 7 GB saying yes, but we expect to get 13 BB saying yes, because either of their children could have been born on a Tuesday and their still have a boy born on a Tuesday.

So we now have 27 fathers who can all make a true statement about their children "I have at least one boy born on a Tuesday". But only 13/27 have BB - not 1/2.


So - we have the same information about each individual in each group - we just arrived at that information a different way. And the probability for individuals from each group having two boys is different, even through we know exactly the same information about them.

Well, actually, we don't have exactly the same information - we also know the information about how we obtained that information, and it's that that makes the difference.