The answer I’m supporting is based on flat priors, not sampling. I’m saying there are two possible Boy/Boy combinations, not one, and therefore it takes up half the probability space, not 1⁄3.
Sampling to the “Boy on Tuesday” problem gives roughly 48% (as per the original article), not 50%.
We are simply told that the man has a boy who was born on tuesday. We aren’t told how he chose that boy, whether he’s older or younger, etc. Therefore we have four possibilites, like I outlined above.
Is my analysis that the possibilities are Boy (Tu) /Girl, Girl / Boy (Tu), Boy (Tu)/Boy, Boy/Boy (Tu) correct?
If so, is not the probability for some combination of Boy/Boy 1/2? If not, why not? I don’t see it.
BTW, contrary to my previous posts, having the information about the boy born on Tuesday is critical because it allows us (and in fact requires us) to distinguish between the two boys.
That was in fact the point of the original article, which I now disagree with significantly less. In fact, I agree with the major premise that the tuesday information pushes the odds of Boy/Boy closer 50%, I just disagree that you can’t reason that it pushes it to exactly 50%.
Is my analysis that the possibilities are Boy (Tu) /Girl, Girl / Boy (Tu), Boy (Tu)/Boy, Boy/Boy (Tu) correct?
No. For any day of the week EXCEPT Tuesday, boy and girl are equivalent. For the case of both children born on Tuesday you have for girls: Boy(tu)/Girl(tu),Girl(tu)/Boy(tu), and for boys: boy(tu)/boy(tu).
That was in fact the point of the original article, which I now disagree with significantly less. In fact, I agree with the major premise that the tuesday information pushes the odds of Boy/Boy closer 50%, I just disagree that you can’t reason that it pushes it to exactly 50%.
This statement leads me to believe you are still confused. Do you agree that if I know a family has two kids, I knock on the door and a boy answers and says “I was born on a Tuesday,” that the probability of the second kid being a girl is 1/2? And in this case, Tuesday is irrelevant? (This the wikipedia called “sampling”)
Do you agree that if, instead, the parents give you the information “one of my two kids is a boy born on a Tuesday”, that this is a different sort of information, information about the set of their children, and not about a specific child?
This statement leads me to believe you are still confused. Do you agree that if I know a family has two kids, I knock on the door and a boy answers and says “I was born on a Tuesday,” that the probability of the second kid being a girl is 1/2? And in this case, Tuesday is irrelevant? (This the wikipedia called “sampling”)
I agree with this.
Do you agree that if, instead, the parents give you the information “one of my two kids is a boy born on a Tuesday”, that this is a different sort of information, information about the set of their children, and not about a specific child?
I agree with this if they said something along the lines of “One and only one of them was born on Tuesday”. If not, I don’t see how the Boy(tu)/Boy(tu) configuration has the same probability as the others, because it’s twice as likely as the other two configurations that that is the configuration they are talking about when they say “One was born on Tuesday”.
Here’s my breakdown with 1000 families, to try to make it clear what I mean:
1000 Families with two children, 750 have boys.
Of the 750, 500 have one boy and one girl. Of these 500, 1⁄7, or roughly 71 have a boy born on Tuesday.
Of the 750, 250 have two boys. Of these 250, 2⁄7, or roughly 71 have a boy born on Tuesday.
71 = 71, so it’s equally likely that there are two boys as there are a boy and a girl.
Having two boys doubles the probability that one boy was born on Tuesday compared to having just one boy.
And I don’t think I’m confused about the sampling, because I didn’t use the sampling reasoning to get my result*, but I’m not super confident about that so if I am just keep giving me numbers and hopefully it will click.
*I mean in the previous post, not specifically this post.
Of these 250, 2⁄7, or roughly 71 have a boy born on Tuesday.
This is wrong. With two boys each with a probability of 1⁄7 to be born on Tuesday, the probability of at least one on a Tuesday isn’t 2⁄7, its 1-(6/7)^2
How can that be? There is a 1⁄7 chance that one of the two is born on Tuesday, and there is a 1⁄7 chance that the other is born on Tuesday. 1⁄7 + 1⁄7 is 2⁄7.
There is also a 1⁄49 chance that both are born on tuesday, but how does that subtract from the other two numbers? It doesn’t change the probability that either of them are born on Tuesday, and both of those probabilities add.
You overcount, the both on Tuesday is overcounted there. Think of it this way- if I have 8 kids do I have a better than 100% probability of having a kid born on Tuesday?
There is a 1/7x6/7 chance the first is born on Tuesday and the second is born on another day. There is a 1/7x6/7 chance the second is born on Tuesday and the first is born on another day. And there is a 1⁄49 chance that both are born on Tuesday.
All together thats 13⁄49. Alternatively, there is a (6/7)^2 chance that both are born not-on-Tuesday, so 1-(6/7)^2 tells you the complementary probability.
I’ve seen that same explanation at least five times and it didn’t click until just now. You can’t distinguish between the two on tuesday, so you can only count it once for the pair.
Which means the article I said was wrong was absolutely right, and if you were told that, say one boy was born on January 17th, the chances of both being born on the same day are 1-(364/365)^2 (ignoring leap years), which gives a final probability of roughly 49.46% that both are boys.
Thanks for your patience!
ETA: I also think I see where I’m going wrong with the terminology—sampling vs not sampling, but I’m not 100% there yet.
The answer I’m supporting is based on flat priors, not sampling. I’m saying there are two possible Boy/Boy combinations, not one, and therefore it takes up half the probability space, not 1⁄3.
Sampling to the “Boy on Tuesday” problem gives roughly 48% (as per the original article), not 50%.
We are simply told that the man has a boy who was born on tuesday. We aren’t told how he chose that boy, whether he’s older or younger, etc. Therefore we have four possibilites, like I outlined above.
Is my analysis that the possibilities are Boy (Tu) /Girl, Girl / Boy (Tu), Boy (Tu)/Boy, Boy/Boy (Tu) correct?
If so, is not the probability for some combination of Boy/Boy 1/2? If not, why not? I don’t see it.
BTW, contrary to my previous posts, having the information about the boy born on Tuesday is critical because it allows us (and in fact requires us) to distinguish between the two boys.
That was in fact the point of the original article, which I now disagree with significantly less. In fact, I agree with the major premise that the tuesday information pushes the odds of Boy/Boy closer 50%, I just disagree that you can’t reason that it pushes it to exactly 50%.
No. For any day of the week EXCEPT Tuesday, boy and girl are equivalent. For the case of both children born on Tuesday you have for girls: Boy(tu)/Girl(tu),Girl(tu)/Boy(tu), and for boys: boy(tu)/boy(tu).
This statement leads me to believe you are still confused. Do you agree that if I know a family has two kids, I knock on the door and a boy answers and says “I was born on a Tuesday,” that the probability of the second kid being a girl is 1/2? And in this case, Tuesday is irrelevant? (This the wikipedia called “sampling”)
Do you agree that if, instead, the parents give you the information “one of my two kids is a boy born on a Tuesday”, that this is a different sort of information, information about the set of their children, and not about a specific child?
I agree with this.
I agree with this if they said something along the lines of “One and only one of them was born on Tuesday”. If not, I don’t see how the Boy(tu)/Boy(tu) configuration has the same probability as the others, because it’s twice as likely as the other two configurations that that is the configuration they are talking about when they say “One was born on Tuesday”.
Here’s my breakdown with 1000 families, to try to make it clear what I mean:
1000 Families with two children, 750 have boys.
Of the 750, 500 have one boy and one girl. Of these 500, 1⁄7, or roughly 71 have a boy born on Tuesday.
Of the 750, 250 have two boys. Of these 250, 2⁄7, or roughly 71 have a boy born on Tuesday.
71 = 71, so it’s equally likely that there are two boys as there are a boy and a girl.
Having two boys doubles the probability that one boy was born on Tuesday compared to having just one boy.
And I don’t think I’m confused about the sampling, because I didn’t use the sampling reasoning to get my result*, but I’m not super confident about that so if I am just keep giving me numbers and hopefully it will click.
*I mean in the previous post, not specifically this post.
This is wrong. With two boys each with a probability of 1⁄7 to be born on Tuesday, the probability of at least one on a Tuesday isn’t 2⁄7, its 1-(6/7)^2
How can that be? There is a 1⁄7 chance that one of the two is born on Tuesday, and there is a 1⁄7 chance that the other is born on Tuesday. 1⁄7 + 1⁄7 is 2⁄7.
There is also a 1⁄49 chance that both are born on tuesday, but how does that subtract from the other two numbers? It doesn’t change the probability that either of them are born on Tuesday, and both of those probabilities add.
The problem is that you’re counting that 1/49th chance twice. Once for the first brother and once for the second.
I see that now, it took a LOT for me to get it for some reason.
You overcount, the both on Tuesday is overcounted there. Think of it this way- if I have 8 kids do I have a better than 100% probability of having a kid born on Tuesday?
There is a 1/7x6/7 chance the first is born on Tuesday and the second is born on another day. There is a 1/7x6/7 chance the second is born on Tuesday and the first is born on another day. And there is a 1⁄49 chance that both are born on Tuesday.
All together thats 13⁄49. Alternatively, there is a (6/7)^2 chance that both are born not-on-Tuesday, so 1-(6/7)^2 tells you the complementary probability.
Wow.
I’ve seen that same explanation at least five times and it didn’t click until just now. You can’t distinguish between the two on tuesday, so you can only count it once for the pair.
Which means the article I said was wrong was absolutely right, and if you were told that, say one boy was born on January 17th, the chances of both being born on the same day are 1-(364/365)^2 (ignoring leap years), which gives a final probability of roughly 49.46% that both are boys.
Thanks for your patience!
ETA: I also think I see where I’m going wrong with the terminology—sampling vs not sampling, but I’m not 100% there yet.