Round 1 of Who Wants to Be a Mathematician had the following math problem:
Bob and Jane have three children. Given that one child is their daughter Mary, what is the probability that Bob and Jane have at least two daughters?
In all such problems we usually make some simplifying assumptions. In this case we assume that gender is binary, the probability of a child being a boy is 1/2, and that identical twins do not exist.
In addition to that, every probability problem needs to specify the distribution of events over which the probability is calculated. This problem doesn’t specify. This is a mistake and a source of confusion. In most problems like this, the assumption is that something is chosen at random. In this type of problem there are two possibilities: a family is chosen at random or a child is chosen at random. And as usual, different choices produce different answers.
The puzzle above is not well-defined, even though this is from a contest run by the American Mathematical Society!
Here are two well-defined versions corresponding to two choices in randomization:
Bob and Jane is a couple picked randomly from couples with three children and at least one daughter. What is the probability that Bob and Jane have at least two daughters?
Mary is a girl picked randomly from a pool of children from families with three children. What is the probability that Mary’s family has at least two daughters?
Now, if you don’t mind, I’m going to throw in my own two cents, that is to say, my own two puzzles.
Harvard researchers study the influence of identical twins on other siblings. For this study they invited random couples with three children, where two of the children are identical twins.
- Bob and Jane is a couple picked randomly from couples in the study with at least one daughter. What is the probability that Bob and Jane have at least two daughters?
- Mary is a girl picked randomly from a pool of children participating in the study. What is the probability that Mary’s family has at least two daughters?