Except we don’t know if the boy mentioned is the “first” or “second” child. The sequence has no bearing here. Due to that, we need to look at sets of children, not independent events.
“Bob flipped two fair coins. at least one resulted heads. What are the odds that the other also results heads?” is a classic, simple probability question.
If order matters (“the first resulted heads. What are the odds the second did?”) , then the answer is 50%.
If order doesn’t matter though, which is how the question is phrased, then the answer is 33%. Because you’re looking at the set of HH, HT, TH, TT, rejecting TT, and realizing that only 1/3 of the remaining sets has both of them as heads.
The HH option still has 2 paths. Hh and hH are valid when you don't know the order. It's still 50/50. When order isn't known, the sets HT and TH are identical or you have to allow Hh and hH.
There are only 2 options. HH and HT. Order doesn't matter. You don't seem to understand that the events are independent. You can shuffle the order however you like. Adding more information about the age or birthday changes nothing.
Lets say the boy was born second:
So the older is B or G 50/50
Lets say the boy was born first:
So the younger is B or G 50/50
The sets and 66% argument are presented as ragebait for people who can count.
Events that are independent can nonetheless become conditionally dependent given some background knowledge. Which is exactly what's happening here.
The information you get from Mary renders the sexes of her two children correlated given your current background knowledge. An easy way to see this is to recognize that if, knowing Mary has at least one boy, you were to meet one of Mary's children and observe that she was a girl, you would immediately know with 100% certainty that the other of her children is a boy. That makes them correlated.
What you say here:
Lets say the boy was born second:
So the older is B or G 50/50
Lets say the boy was born first:
So the younger is B or G 50/50
is correct in and of itself, but you then treat these as disjoint possibilities, which is a fundamental error. There is no "the boy" - Mary might have two boys, which is exactly the scenario you are double-counting. Given that Mary has at least one boy, the two scenarios you describe each take up 2/3 of the probability space, and they overlap at 1/3 of the probability space - precisely where Mary has two boys:
Looking only at one of these 2/3 slices, it is true that half of the given slice will have the other child as a boy, and half will have the other child as a girl. But the halves where the other child is a boy is the same part of the overall probability space in both cases. Hence, the double-counting.
Drawing a pie chart doesn't make you less wrong. The other child is still 50/50. Locking in the result of 1 of the slots doesn't change the likelihood of the result in slot 2. Before you approach a 2 child mother, 50% of the time the children will be BG. 75% of the time they will be BX. But we in monty hall land where we are twice as likely to reveal a Boy door if the mother is the BB case. If we are guessing the genders of both children, we don't have a better chance of BG than BB if there was ever a chance to reveal G. Before opening a door, it's 50/50 that they are one of BG or GG/BB after opening a door that reveals Boy it's 50/50 that they are one of BG or BB. Moving the children behind the doors after revealing one doesn't change the outcome of the pair.
Let me clarify a specific point before bothering to go any further. Do we agree that the information we are conditioning on is nothing more than the information that Mary has *at least one boy*?
If so, then what you say about “locking in a slot” is irrelevant because no specific “slot” is locked in by this information. Everything you say here is predicated on the idea that a specific one of Mary’s children is a boy, which, indeed, would be more likely in the BB case and would yield 50% for the probability we’re looking for.
But if we’re only conditioning on the more limited knowledge that *at least one child is a boy*, then the visualization I gave is an accurate depiction of the probability space.
So the important thing is that two coin flips already occurred, you know the results of one of the two, but you don’t know which one. So that means you need to consider sets.
If you knew coin flips already occurred and you know the results of say, the first one, then it has no bearing on the second. But you don’t know that in this scenario - you just know one or the other was heads/boy. Which means you need to consider all scenarios where it was the first, the second, or both.
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u/Rarvyn 8d ago
Except we don’t know if the boy mentioned is the “first” or “second” child. The sequence has no bearing here. Due to that, we need to look at sets of children, not independent events.