Another probability paradox – was first introduced by Martin Gardner in 1959. Let me introduce my version of the questions.
1) Mr X has two children. The first one is a girl, and what is the probability that both the children are girls?
2) Mr Y has two children. At least one of them is a girl, and what is the probability that both the children are girls?
No Two on the First
There are many ways to solve it, the simplest first. If the first one is a girl, all you have to get is the chance that the second child is a girl. Since the gender of the second child doesn’t depend on the first, the probability is 1/2. So the answer is 1/2.
The is another way, write down all the possibilities. BB, BG, GB, and GG; B represents boy, G means girl. Since the first person is a girl, the first two options go away, and the answer to choose 1 out of 2 for a GG = 1/2.
The Second will Psych You Up
The second problem is where the controversy is. Let’s take the second approach to solve it. As before, the four possibilities are BB, BG, GB, GG. Since at least one is a girl, BB goes away, and your chance of getting GG out of 3 options is 1/3.
Simple? But I do not want to agree with that. What was the difference if the question was like this: Mr Y has two children. At least one of them is a girl, and what is the probability that the other is also a girl? For all practical purposes, both the statements of problem 2 mean the same, perhaps not for mathematicians (by the way, this can be the fallacy to counter!). If at least one is a girl, the other can only be one of the two – which is 1/2. So?
What about this: if the prior information says at least one of them is a girl, you can not have BG and GB as two options (like permutation vs combination)?. So you have to look for GG from two options GG and BG/GB 😜.
Wiki page on Boy or Girl Paradox
Online Discussion on the problem
