Boy/girl riddle (conditional probability)

[SlowThinker]

I've found this video about conditional probability:

All steps look correctly, but the result does not make any sense.
I'm ok with the part about frogs, but not so with the boy/girl computation.

To sum it up:
1) I have two children and at least one of them is a boy. What is a probability I have a girl?
Answer: 2/3

2) I have two children and at least one of them is a boy who was born on Tuesday. What is the probability I have a girl?
Answer: ~52%

It does not make any sense... to me anyway.
Whatever day of week I say, it tells you nothing about the boy/girl status, does it?
If I said the boy was born on January 1, 2001, would it shift the probability even closer to 50%? How is that possible?
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If you don't want to watch the video:
1) out of the 4 equally probable possibilities $BB$, $BG$, $GB$, $GG$, the $GG$ is crossed out, leaving 2 of 3 cases containing G.

2) out of the 14^2 equally probable cases $B_{Monday}B_{Monday}$ .. $G_{Sunday}G_{Sunday}$, we have:
7 cases $B_{Tuesday}B_{any}$,
6 cases $B_{not\ Tuesday}B_{Tuesday}$
7 cases $B_{Tuesday}G_{any}$,
7 cases $G_{any}B_{Tuesday}$.
So, to have a G, we have (7+7)/(7+6+7+7)=14/27 ~ 52%.

 

[PeroK]

This has been done to death. Ultimately it depends how you obtain the information. If you ask two direct questions of someone:

Do you have exactly two children?

Do you have at least one boy born on a Tuesday?

Then, the second question is more likely to be answered yes by a parent with two boys.

Whereas, if you simply ask.

Do you have at least one boy?

Then that will be answered yes equally by a parent with one boy or two.

However, if someone simply volunteers this information, there is no logical way to deduce why they are saying it. Nor to deduce how likely it is that they have one or two boys.

 

[PeroK]

Just to elaborate on one point. Suppose you ask the questions as follows:

Do you have exactly two children?

Do you have at least one boy?

Then, slightly awkwardly, think of your boy if you have one boy and pick one of your boys at random if you have two boys. Tell me his birthday.

Then, if the answer is Tuesday, it makes no difference to the original 2/3 answer because however many boys they have they will simply give a day of the week.

Again, the key point is that the questioner has to introduce the day of the week to affect the sample space.

 

[PeroK]

If I said the boy was born on January 1, 2001, would it shift the probability even closer to 50%? How is that possible?

If you go around asking this question, then that is exactly what it does. Very few parents will answer yes to this question. But overall parents with two boys are twice as likely to answer yes. So you narrow your sample space to be equally likely BB and BG.

Again, the provisos above about how this information is obtained apply.

 

[SlowThinker]

I find it confusing that of the sentence "I have a boy born on Tuesday", the boy part gets factored in, and the Tuesday part gets ignored.

So...
Scenario 1:
"I have 2 children and at least one of them is a boy".
This says nothing at all about the other child, because you don't know my motivation for saying that. So it's 1/2 for a girl.

Scenario 2:
"I have 2 children."
Now the probability I have a girl is 3/4.

Q: "Do you have a boy?"
A: "Yes."
Now the probability I have a girl is 2/3.

If my explanation of Scenario 1 is correct, I think it all makes sense now.

 

[PeroK]

There are two problems with scenario 1. First, you're not talking about a specific child. If you say my older child is a boy, then it's 1/2 that the other is a boy or girl. But, if you just say I have at least one boy, then the answer is still 2/3. Sort of ...

Because the real problem is why you would say this? Let's expand the sample space to all parents with two children. Some parents might say nothing about their children and talk about the weather. Some parents might say "I've got two girls". Eventually someone might say "I have at least one boy". Why would anyone say that? You don't have logical probabilistic criteria on which to work. Maybe only a parent with one boy and one girl would ever say that. You simply don't know why they said that rather than something else.

You can of course take this statement at face value and assume that it's just a random statement from a random family. Then everything works out as in the video.

But, it's cleaner probabilistically if the information given is an answer to a direct question. Then you know exactly where you stand probabilistically.

 

[FactChecker]

I find it confusing that of the sentence "I have a boy born on Tuesday", the boy part gets factored in, and the Tuesday part gets ignored.

So...
Scenario 1:
"I have 2 children and at least one of them is a boy".
This says nothing at all about the other child, because you don't know my motivation for saying that. So it's 1/2 for a girl.

Scenario 2:
"I have 2 children."
Now the probability I have a girl is 3/4.

Q: "Do you have a boy?"
A: "Yes."
Now the probability I have a girl is 2/3.

If my explanation of Scenario 1 is correct, I think it all makes sense now.

That sounds wrong. Knowing there is at least one boy increases the odds that there are two boys. The possibility of 2 girls has been thrown out, so that reduced the odds of the other child being a girl. The girl odds started at 1/2, but after throwing out the two-girl option, they are reduced.

Regarding the Tuesday part, the problem statement has not been presented in a way that you can answer precisely. Consider these two different problems:
1) Suppose the problem had said that you would only know there was a boy if he was born on Tuesday. Otherwise, you would not know. Then the more boys there are, the more likely that one would be born on Tuesday and you would know it. So the information that there is a Tuesday boy definitely increases the odds that there are two boys.

2) Suppose the problem had said that you know there was a boy and you asked him and found out that he was born on Tuesday. Then the Tuesday information has no effect on the probabilities. He was going to be born on some day. It might as well be Tuesday. The Tuesday part did not change the odds that you found out there was a boy. So that information does not give you any more clues as to how many boys there are.

 

[SlowThinker]

That sounds wrong. Knowing there is at least one boy increases the odds that there are two boys. The possibility of 2 girls has been thrown out, so that reduced the odds of the other child being a girl. The girl odds started at 1/2, but after throwing out the two-girl option, they are reduced.

I see this riddle is ambiguous.
Imagine we meet:
I: "I have 2 children."
You: "When were they born?"
I: "Peter was born on a Tuesday."

Now you know I have a boy, and he was born on Tuesday.
If I preferred to talk about boys, you'd guess the chance of the other child being a girl is 2/3.
If I preferred to talk about the older child (or had no preference), you'd guess 1/2.
If I preferred weekdays near the beginning of the week, you'd guess 1/2 again but might conclude something about the day of birth of the other child.
To arrive at the 52%, the conversation would have to be different ("Do you have a boy born on Tuesday?" - "Yes").

So I have to be careful with the exact wording and assumptions, or I could arrive at several different conclusions.