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.
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.