by onecommentman | 2025-08-29 01:17:47+00:00
When I first heard of the Monty Hall problem, I assumed the naive answer was true, then I thought more and drew up the decision tree and thought the “correct” answer was right. Now I think it is an underspecified problem and literally any probability can be assigned to the result. There are no bad answers because it is a bad(ly stated) problem.

“Here's the problem: a family has two children. You're told that at least one of them is a girl. What's the probability both are girls?”

— This is the complete statement of the problem! Everything else is an assumption that may or may not be correct. And is certainly not necessarily a complete set of underlying assumptions relevant to the problem statement.

“Assume that the probability of having a girl or boy is 50% and that the birth order has no effect on the probability. Assume the family is selected at random because they have at least one girl.”

— This is not a part of the statement of the problem! These are a subset of assumptions that can choose to accept, or not. As a modeler or decision analyst you have to make that distinction. Eh, let’s accept them, for the time being. We’ll even assume the narrator is honest, which isn’t a stated assumption.

But let’s add to that list of assumptions. The narrator telling you that one of them is a girl gets all winnings from bets on the outcome of the unknown gender of the “other child” and wants those winnings. The narrator knows that a probability tree analysis of the problem, with perhaps unwarranted assumptions of independence and prior probabilities, will lead to an assignment of 1/3 probability for the other sibling being a girl, and knows you know that result and believes you want to win. [A valid credible interpretation, not misinterpretation, of the original problem statement.]

“What do you think the probability is that both children are girls?”

— Let’s make this question more actionable. “Should you take the even odds bet on both children being girls made by the narrator?. $100 - if they are both girls, the narrator wins $100 and you lose $100; if they aren’t, you win $100 and the narrator loses $100. The narrator and you want to win the money.”

The answer to this question, which seemingly follows from the question of probabilities to be “yes”, is, in fact, “no” - under the additional valid, and quite credible, assumptions made. Because you will only be presented sets of two-girl pairs by the narrator. Let’s assume the “assumptions” are actually correct, and the families will be indeed selected at random, and in the general population there is a 50-50 mix of boys and girls. There is nothing, even in the “assumptions”, that precludes the narrator from preselecting and only presenting two-girl pairs to you, thus always winning when you believe and follow the 1/3 two-girl result.

The statement of the problem, and only the statement of the problem, underspecified as it is, leads to a whole suite of possibly correct answers. The problem is the territory, the problem statement and assumptions are the map.

None of these maps are the territory, necessarily. The probability tree answer is just as sloppy, from a decision analyst perspective, as the naive answer.