one of the more intriguing applications of conditional probability." Of course, the actual probability values do not matter; the purpose of the paradox is to demonstrate seemingly contradictory logic, not actual birth rates. Information about the child Suppose we were told not only that Mr. Smith has two children, and one of them is a boy, but also that the boy was born on a Tuesday: does this change the previous analyses? Again, the answer depends on how this information was presented – what kind of selection process produced this knowledge. Following the tradition of the problem, suppose that in the population of two-child families, the sex of the two children is independent of one another, equally likely boy or girl, and that the birth date of each child is independent of the other child. The chance of being born on any given day of the week is . From Bayes' Theorem that the probability of two boys, given that one boy was born on a Tuesday is given by: Assume that the probability of being born on a Tuesday is ε = which will be set after arriving at the general solution. The second factor in the numerator is simply , the probability of having two boys. The first term in the numerator is the probability of at least one boy born on Tuesday, given that the family has two boys, or (one minus the probability that neither boy is born on Tuesday). For the denominator, let us decompose:. Each term is weighted with probability . The first term is already known by the previous remark, the last term is 0 (there are no boys). and is ε, there is one and only one boy, thus he has ε chance of being born on Tuesday. Therefore, the full equation is: For , this reduces to If ε is now set to , the probability becomes , or about 0.48. In fact, as ε approaches 0, the total probability goes to , which is the answer expected when one child is sampled (e.g. the oldest child is a boy) and is thus removed from the pool of possible children. In other words, as more and more details about the boy child are given (for instance: born on January 1), the chance that the other child is a girl approaches one half. It seems that quite irrelevant information was introduced, yet