I was thinking about conditional probability which refers to the chance of outcome A if condition B is true. We use conditional probability in medicine when we try to figure out how the results of a research study will apply to a particular patient.

Conditional probabilities are often tricky to figure out as they can be extremely counter-intuitive. What seems to happen is that any information contained in the condition can influence the probability, even if the information seems to have nothing to do with the question.

A famous conditional probability problem is the “Monty Hall” problem, which became famous when Marilyn vos Savant gave the correct answer. It was very counter-intuitive and she and received boatloads of critical mail a lot of it from people who claimed to have Ph.D.’s. Unfortunately for them, she was correct.

Here is another example of conditional probability weirdness.

I put the condition in bold. Notice how the condition changes the probability changes.

John and Susan have two children who are not twins. The probability that any child is a girl is 1/2.

What is the probability that both are girls? (No condition)

Answer: 1/4

What is the probability that both are girls **if the older child is a girl**?

Answer: 1/2

What is the probability that both are girls **if one child is a girl**?

Answer: 1/3

What is the probability that both are girls **if one child is a girl with curly hair**? Now curly hair should have nothing to do with the probability of the sex of the children. So we would expect the answer to be 1/3 just like in the situation above where one child is a girl.

But …

Answer: The probability that both children are girls if one child is a girl with curly har is between 1/3 and 1/2 depending on the frequency of curly hair in the population.

That is completely bizarre. It is so counter-intuitive that even after doing the math to get the result I wrote a program in Matlab to check and make sure it was correct. I’ll triple check my work and post if I find an error.