This article presents "The Two Children Problem," published by Martin Gardner, who wrote a famous and widely-read math puzzle column in the magazine "Scientific American," and a problem presented by puzzler Gary Foshee. This paper explains the paradox of Problems 2 and 3 and many other variations of the theme. Then the authors look at the solutions for problems 2 and 3 and their generalizations and compare the results. Finally, the authors use these problems to draw attention to the importance of both conditional probability and sample spaces for teaching probability at all levels. Paradoxes are a wonderful way to get teachers, students, and their parents thinking hard about mathematics and solving problems and variations. Foshee's puzzle is a new contribution which can be used in schools.