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# Surprising Probabilities: When Intuition Struggles

## BOYS, GIRLS, ACES, AND COLORED CARDS

Probability is notorious for problems that are easy to state but whose solutions can lead to confusion, heated disputes, and hurling of insults. Even though I have never personally witnessed a discussion of a probability problem turn into a fist fight, I would not consider it to be impossible. Most often the confusion stems from problems that are not well formulated and several different interpretations are possible. The high emotional level may have to do with the fact that probability problems often relate to everyday phenomena. It is in this way different from many branches of mathematics because you can often understand the problem and suggest solutions without having any formal training in probability. Let us start with a standard problem of this kind.

Your Aunt Jane calls again, this time to tell you that her new neighbors have two children and that at least one is a boy. What is the probability that the other is also a boy?

You discuss this problem with Albert and Betsy down at the pub. Your opinion is that one child is a boy and the gender of the other child is independent of this fact; thus, the probability is simply 1/2. Albert, on the other hand, wants to be more methodical. He lists the sample space BB, BG, GB, and GG, and notes that the outcome GG is impossible. As we know that at least one child is a boy, we have either BG, GB, or BB, and as one out of the three has another boy, ...