All of this by way of introducing a nice puzzler on the topic. The source for this is this blog post. If I told you that I had two children, that the first was male, and ask what you thought the probability of the second being male is, we could write it in conditional probability notation like this:
Pr[2nd child is male | there are two children, and the 1st child is male]The vertical bar is read as 'given that'. In real life, this isn't so simple, since gender is determined by the chromosomal contributions of the male, and we can't necessarily assume it's 50-50 (some males have only male children, for example). Also, slightly mre males than females are born because mortality rates aren't the same. But assume these complications away. Assume it's 50%.
What if we made the question more general? Find:
Pr[both children are male | there are two children, and at least one child is male]It looks similar, but it's subtly and significantly different. The probability is not 50% anymore. Now it gets really strange. Find:
Pr[both children are male | there are two children, and at least one child is male born on a Tuesday]The probability is now different from the previous one, which seems to confound all common sense. How can such "irrelevant" information as the day of week of birth have any impact on whether the other child is a male or not? A full discussion of the problem and solution can be found on the post. And there's an interesting discussion on reddit, where I found the problem.