Conditional Probability · Conditional Probability
Lesson 6
The wording of this problem is confusing. Notice the subtle difference with the following wording:
A family has two children, the younger of whom is a boy. What is the probability that the older one is also a boy?In this wording, it is clear that the probability is \(1/2\) (we, of course, assume that the probability of each gender is \(1/2\) and that the gender of the second child does not depend on the gender of the first child; whether this is true in real life is not important right now). But in the problem, it is stated that one of the two children is a boy. Thus, the original probability space \[\{\text{GG}, \text{GB}, \text{BG}, \text{BB}\}\] (here, the first letter denotes the gender of the older child, whereas the second letter is the gender of the younger child) is reduced to the following: \[\{\text{GB}, \text{BG}, \text{BB}\} \ .\] Thus, the probability that the couple has two boys is \(1/3\). This problem is known as the Boy or Girl Paradox. The article, in particular, discusses that the question should be formulated more precisely to expect a clear answer.