It’s so counter-intuitive my head is going to explode.
Here’s the paradox for the uninitiated:If I say, “I have 2 kids, at least one of which is a girl.” What is the probability that my other kid is a girl? The answer is 33.33%.
>*Intuitively, most of us would think the answer is 50%. But it isn’t. I implore you to read more about the problem.*
Then, if I say, “I have 2 kids, at least one of which is a girl, whose name is Julie.” What is the probability that my other kid is a girl? The answer is 50%.
>*The bewildering thing is the elephant in the room. Obviously. How does giving her a name change the probability?*
>
>*Apparently, if I said, “I have 2 kids, at least one of which is a girl, whose name is …” The probability that the other kid is a girl* ***IS STILL 33.33%.*** *Until the name is uttered, the probability remains 33.33%. Mind-boggling.*
And now, if I say, “I have 2 kids, at least one of which is a girl, who was born on Tuesday.” What is the probability that my other kid is a girl? The answer is 13/27.
>*I give up.*
Can someone explain this brain-melting paradox to me, please?
In: 4
I completely disagree with the claim of 33.33% in the first place.
Consider the gambler’s fallacy, which we know is wrong, which goes like this:
“I flipped this coin and got 2 heads in a row therefore my next flip is more likely to be a tail because 3 heads in a row is only 1/8 likely. I’ve got at 7/8 chance for a tail now.”
That is the fallacy of treating outcomes that have already been set in stone (in the past) as if they were still open possibilities that are part of the probability prediction about the future.
The gambler in the fallacy is thinking of these 8 outcomes:
– HHH
– HHt
– HtH
– Htt
– tHH
– tHt
– tTH
– TTT
And forgetting that we already eliminated 6 of them by setting the first two flips in stone as both heads, so the remaining possibilities are just these:
– HHH
– HHt
– ~~HtH~~ ruled out already
– ~~Htt~~ ruled out already
– ~~tHH~~ ruled out already
– ~~tHt~~ ruled out already
– ~~tTH~~ ruled out already
– ~~TTT~~ ruled out already
Thus the chance on the next flip is now the same as it was on any normal flip. It’s 50% between the remaining 2 options.
And that fallacy is exactly what the answer “33.33%” is incorrectly doing here. It starts with only telling you there are 2 kids, which means these 4 possible open ‘future’ outcomes, like so: (“Future” as in you don’t know the information yet.)
– BB
– BG
– GB
– GG
Then it tells you it picked one and identified it as a girl, then *pretends* that only eliminated the BB outcome leaving 3 left, but it didn’t. It eliminated BOTH BB and BG because by assigning one of them making it no longer unknown, it essentially put it “in the past” in the gambler’s fallacy. It’s already been set in stone. That *creates* an implicit ordering between the two kids, in the sense that one was identified in the past, a few seconds ago, and the other will be identified in the future, not yet. And you must never use the events already set in stone in the past as if they were part of a probability calculation.
By pretending that both BG and GB are still possible, the “33.33%” answer is violating that rule. They are not both still possible. You pointed at one of the kids and said “that one is not a boy”. That already happened. That’s already a past event. So it looks like this:
– ~~BB~~ ruled out
– ~~BG~~ ruled out
– GB
– GG
The notion that this ordering that differentiates BG from GB ONLY happens if you named the one you pick is wrong. The fact that one kid has already had their sex disclosed while the other has not *IS THE ORDERING* that lets you call the already-disclosed one the lefthand letter and the not-yet-disclosed one the righthand letter, of the pair of letters.
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