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
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.
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