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
This problem is actually a notorious example of how it can be difficult to assign meaningful probabilities to everyday statements, at least so long as those statements leave room for some unorthodox interpretations of the information provided.
The first question gets us into the spirit. If it had asked about families where the *oldest* daughter was a girl, then the probability of a second girl would be the intuitive 1/2. This is because the information about one specific child is not informative about the other. However, we’re instead told just that *one* of the children is a girl, so we have to consider all possible family formations (BB, BG, GB, and GG), restrict to the families that satisfy the condition (BG, GB, GG), and calculate the percentage that have a second girl. As other users have pointed out here, that’s 1/3.
But then the second question, in a sense, takes things “too far”. We intuitively think that the information that the girl’s name is Julie is incidental to the procedure just discussed. We could have picked a family with one girl that doesn’t have a daughter named Julie. However, the person discussing the paradox isn’t treating it that way. For them, having a daughter named Julie is necessary to be a selected family. That requirement actually changes the set of families we could draw from because families with two girls get two chances to have a girl named Julie. The population being sampled from is thus BG(j), G(j)B, G(j)G, and GG(j) – where (j) indicates that the Girl is named Julie). Half of those families have two girls. The weekday of birth works similarly – it treats the girl-born-on-Tuesday condition as essential to being sampled, giving families with two girls more chances to be sampled. The math is just more annoying.
Editing to address a few common misconceptions I’m seeing in the comments:
* I wrote possible families in a way where order matters because it was an easy way to keep an event space where all events have equal probability. An alternative way to do the first question would be to say there are three possibilities: {2 boys, 1 girl and 1 boy, and 2 boys}. You just have to keep in mind that the middle possibility is twice as likely as either extreme. This is because the families themselves are formed by independent draws from a 50-50 gender pool, regardless of what subset we’re talking about now or later in the question.
* Likewise, the probability of finding a family with any daughter named Julie is quite low, so the probability of e.g. GG(j) is much much lower than BB. However, because every possible family considered in that part has exactly one girl named Julie, you’re just multiplying all 4 by the marginal probability that any girl is named Julie, so it cancels out in any comparisons and you don’t even need to know what it is.
* There are of course many simplifying assumptions being made to keep focus on the probability puzzle here.
* Yes, a family could name BOTH their daughters Julie. To properly take this into account, you would have to actually know the probability that any given girl is named Julie.
* Sex ratios are naturally skewed just a hair towards females and sometimes unnaturally skewed towards males, so treating it like a coin flip isn’t entirely accurate.
* (Nobody actually caught this one, but it’s my favorite): Among 2-child families who have stopped having kids (who will be a subset of the families you’ll draw from and hence will skew results relative to the mathematical ideal), MORE than 50% will be one boy and one girl. It’s a documented phenomenon that families with two children of the same sex are more likely to have a third due to a preference for having at least one boy and at least one girl. This doesn’t sway THAT many families, but it’s enough to make a detectable difference.
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