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
There are four equally likely gender configurations of families that have two children: male/male (1), male/female (2), female/male (3), female/female (4). The statement that at least one is a girl eliminates family #1. So you’re picking randomly from the three other families. Only in family 4 is the other child a girl. So one in three odds.
This is a variant of the Monty Hall problem. You have an unknown probability and some information. So, if you just say “I have two children”, there are four options:
* two boys
* older girl, younger boy
* older boy, younger girl
* two girls.
When you add the fact that one is a girl, you have only eliminated the first possibility. However, that fact says nothing about which of the other three possibilities it is. In two of the three cases, the other child is a boy, so what’s left is a 1 in 3 chance that the other child is a girl.
Giving a name or a birthday adds more information. There are 27 possibilities for one child being born on a Tuesday, 13 of which are the pairing of two girls.
[Chart](https://www.jesperjuul.net/ludologist/wp-content/uploads/2010/06/fullGrid.png)
The name is even more unique, as there are a near infinite number of names the children could have. Technically the probability of a second girl is 49.99999…999%, but you can round to 50 because there are only so many people who would name their children the same thing.
*George Foreman has entered the chat*
pl487 answered the first part.
As for the Julie part by saying one of the children is Julie you no longer have this distribution: MM MF FM FF. Instead you have Julie/m Julie/f OR m/Julie f/Julie.
By knowing that on child is not just a girl, but a specific girl you have a different distribution of possibilities.
For the scenario “I have 2 kids, at least one of which is a girl, whose name is Julie”, start again with the basic possibilities:
Boy/Boy. Boy/Girl. Girl/Boy. Girl/Girl.
Now, *the parent knows* which of the children (the older or the younger) is Julie. Let’s say that Julie is the younger child.
So now the possibilities become:
Boy/Boy. Boy/Julie. Girl/Boy. Girl/Julie.
There are only two possibilities in which Julie is the younger child, and of those possibilities the chance of both children being girls is 1 in 2, or 50%.
We could also have assumed that Julie is the older child, and the result is the same.
This “paradox” depends on a linguistic trick where by naming the child you are changing one interpretation of the phrase’s meaning and considering a different situation.
Consider the first question: “I have 2 kids, at least one of which is a girl. What is the probability that my other kid is a girl?” There are four different possible ways the children could be born:
Girl and Girl
Girl and Boy
Boy and Girl
Boy and Boy
We can eliminate the last option from consideration because one of them isn’t a girl, meaning we only have the first three. Of those three only the first has the other child being a girl, so the probability is 33.33%
It is important to note that the situation of “Girl and Boy” is being counted as distinct to “Boy and Girl” even though they both equate to there being one boy and one girl. This is because while the end result is the same the probability of having both boys or both girls is not the same as one of each. It is this difference which the “paradox” is exploiting with ambiguous phrasing.
Consider the second question: “I have 2 kids, at least one of which is a girl, whose name is Julie.”
In this situation it is being interpreted that we are picking a child from an existing pair of children, which combines the “Boy and Girl” and “Girl and Boy” possibilities. So instead we have these options:
Girl and Julie
Boy and Julie
Boy and Boy
Again we can eliminate the last option from consideration because we know one isn’t Julie, meaning we only have two options left. Therefore it is now 50% that the other child is a girl.
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However, I would argue this is an improper twisting of linguistics and probability. We already established that the chances of having both a boy and a girl is equal to the chances of having both children the same sex. Therefore we would expect that there would be twice as many families out there with Julie and a boy compared to Julie and a girl, even though there are just a pair of options. Just because there are two options doesn’t mean they are equally probable.
Let’s analyze the possible scenarios within the sample space: the family can have four different formats, listed as “older child; younger child”.
* Boy; boy
* Boy; girl
* Girl; girl
* Girl; boy
We can conclude that the first scenario is not possible since we know that at least one of them is a girl. Therefore, the probability of having two girls is 1/3.
When we assign a name to one of the girls, it affects the probability because it provides more ways to distinguish between the two sisters. If we rephrase the sample space as follows:
* Julie; girl (not Julie)
* Julie; boy
* Boy; Julie
* Girl (not Julie); Julie
* Boy; boy
Once again, it is clear that scenario “boy; boy” is not possible. However, this time, there are two outcomes (1 and 4) that correspond to outcome 3 in the previous question. Therefore, the probability of having two daughters is 1/2.
The example for when they are born on Tuesday is slightly more complicated. However, let’s write out all the ways you can have two children, with one being a girl born on a Monday. Here are the number of each combination of Boy (B), Girl born on a Monday (GM) or Girl born not on a Monday (GNM) you would expect if you have 196 pairs:
* (GM)B 7
* (GNM)B 42
* B(GM) 7
* B(GNM) 42
* (GM)(GM) 1
* (GNM)(GM) 6
* (GM)(GNM) 6
* (GNM)(GNM) 36
* BB 49
If you count them up, you get 27 scenarios with one girl born on a Monday, and of these 13 have two girls, giving you 13/27 as your odds.
The reason it appears paradoxical, but isn’t, is that the more information you provide about a child, the smaller the likelihood of there being two children like that, and so the closer, the more possible combinations of the two children there are.
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.
I’d argue the permutations are boy/Boy, boy/Boy, girl/boy, boy/girl, girl/girl, and girl/girl.
The posters are including the permutation for the girl/boy and boy/girl but removing 1 of the 2 permutations of the same sex pairs.
It’s the difference between past and future events.
If the question was “I have one child that is a girl, what is the probability that my next child will be a girl”, the answer would be 50%.
You already have 2 children so we’re trying to guess what happened in the past. The probability of a correct guess changes based on the information we have about the things that already happened.
The 33% answer is wrong because saying that at least one of them is a girl introduces new information. In mathematical notation it is a conditional probability e.g P(A|B).
The way to think about this is:
First child is a girl for sure. Second child may be boy or girl. Since there are only 2 options the probability is 50%.
Again, boy girl is same as girl boy. The order doesn’t matter because the question doesn’t imply that order matters.
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