Two probability questions for nerds

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Duke of Marmalade

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Q1 A lady tells you she has 2 children. You ask has she a boy and she says Yes. What is the probability she has 2 boys, given that one of them is a boy?

Q2 Same lady. You ask her has she a boy born on a Tuesday and she says Yes. What is the probability she has 2 boys?

You may assume that it is 50% that a child is a boy and that if the eldest is a boy it is still 50% that the youngest is a boy i.e. independent events.
You may also assume that there are equal chances for any day of the week that a child is born.
 
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My initial answer was the same as Protocol.
There are three possible combinations* so at the outset the odds of two boys is 33.3%
We now know that one of the children is a boy so presumably that alters the odds as one of those combinations is now impossible.
I still would have said 50% are the odds at this point! However given that it is not then my next guess is 66.6% but I'm not even sure of the logic behind this!
The answer for Q1 and Q2 has to be the same doesn't it?

* Assuming the baby hasn't yet identified themselves as non binary!
 
I don't understand probability so I'll read with interest if not understanding.
 
Ceist Beag said:
My initial answer was the same as Protocol.
There are three possible combinations* so at the outset the odds of two boys is 33.3%
We now know that one of the children is a boy so presumably that alters the odds as one of those combinations is now impossible.
I still would have said 50% are the odds at this point! However given that it is not then my next guess is 66.6% but I'm not even sure of the logic behind this!
The answer for Q1 and Q2 has to be the same doesn't it?

* Assuming the baby hasn't yet identified themselves as non binary!
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Ha! Ha!
Your answers are not right but it is time for a clue. You are right that at the outset there are 3 possibilities but they are not equally likely.
And believe it or not the answers to the questions are not the same.
 
For Q1, at the point where you know there are two children, and one is a boy, does it not boil down to the 50/50 on the gender of the second?

Duke of Marmalade said:
You may assume that it is 50% that a child is a boy and that if the eldest is a boy it is still 50% that the youngest is a boy.
Click to expand...
 
Leo said:
For Q1, at the point where you know there are two children, and one is a boy, does it not boil down to the 50/50 on the gender of the second?
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I didn't say it was the first that was a boy. Maybe it was the second or maybe it was both.
 
Leo said:
For Q1, at the point where you know there are two children, and one is a boy, does it not boil down to the 50/50 on the gender of the second?
Click to expand...

I don't think so. The boy could be the eldest or the youngest so alternatives would be boy/boy, boy/girl or girl/boy. Duke is saying that they are not equally likely so thats the bit I don't get.

No idea on the second one...
 
Sunny said:
I don't think so. The boy could be the eldest or the youngest so alternatives would be boy/boy, boy/girl or girl/boy. Duke is saying that they are not equally likely so thats the bit I don't get.

No idea on the second one...
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Ahhh! You're getting close. I did not say that those three are not equally likely. Read my reply to @Ceist Beag carefully.
 
Q1 is 25%. She can have either BB, BG, GB, or GG.
No idea on Q2, but I'd say it's also 25%, in that the sex of a child is not conditional on the day of the week a sibling was born.
 
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Sunny said:
Damn you Duke, I have work to do!

Ok, I am going with 1/3 chance that the other child is a boy....
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Yep! :) 1/3rd is the answer.

Following @Ceist Beag 's approach we argue as follows.
From the outset there are 3 possibilities
2 boys: 25% chance (50% the eldest x 50% the youngest)
2 girls: 25% chance
1 boy, 1 girl: 50% chance (because they can come in either order or simply because this must be the remaining 50%)

The second possibility is then ruled out and we are left with the other two in the same proportion as at outset i.e. 25 to 50 which gives a 1/3rd chance of 2 boys.

Now Q2 is a different kettle of fish.
 
Sunny said:
Damn you Duke, I have work to do!

Ok, I am going with 1/3 chance that the other child is a boy....
Click to expand...
In my experience with hard work and discipline you can overcome your innate urge to learn things and better yourself.
 
Duke of Marmalade said:
Yep! :) 1/3rd is the answer.

Following @Ceist Beag 's approach we argue as follows.
From the outset there are 3 possibilities
2 boys: 25% chance (50% the eldest x 50% the youngest)
2 girls: 25% chance
1 boy, 1 girl: 50% chance (because they can come in either order or simply because this must be the remaining 50%)

The second possibility is then ruled out and we are left with the other two in the same proportion as at outset i.e. 25 to 50 which gives a 1/3rd chance of 2 boys.

Now Q2 is a different kettle of fish.
Click to expand...

Yeah I will leave Q2 to others! Nice idea to throw them out there...
 
Duke of Marmalade said:
Yep! :) 1/3rd is the answer.

Following @Ceist Beag 's approach we argue as follows.
From the outset there are 3 possibilities
2 boys: 25% chance (50% the eldest x 50% the youngest)
2 girls: 25% chance
1 boy, 1 girl: 50% chance (because they can come in either order or simply because this must be the remaining 50%)

The second possibility is then ruled out and we are left with the other two in the same proportion as at outset i.e. 25 to 50 which gives a 1/3rd chance of 2 boys.

Now Q2 is a different kettle of fish.
Click to expand...
No. Odds are the inverse of probabilities, not possibilities. If 2 boys are 25% chance the odds are 4 to 1, i.e 25%, i.e. 1/4.
 
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