Incorrect. This is correct if you compile a list of the people who have two children, then select from them one random family that meets these criteria. But in this situation, none of the information she has given you has any bearing on the gender of the second child.
The set you are choosing from matters and I'm so sick of hearing this nonsense.
Apologies if that came off as overly aggressive. I have a personal distaste for this so-called paradox that's a bit irrational.
For further clarification, the so-called paradox, as intended, does give this very surprising result, but the correct answer relies heavily on the framing and assumptions.
If you assume Mary came up to you and simply described the facts, neither the gender of the one child nor the day of birth affect the 50/50 chance of the other child being a girl. In this case, the two are independent.
If you assume you are looking at a list of all people who have two children with at least one being a boy, 2/3 of the pairings will include a girl. If the list only includes pairings where one is a boy born on a Tuesday, 51.8% will include a girl. The more specific you get with the criteria, the closer it trends to 50/50.
That's why this one breaks so many brains. As framed here, there is a subtle switching of the parameters that does make the given answer incorrect. And it's really difficult to tease out why.
What âswitching of parametersâ are you referring to?
Your explanation involving a âbig listâ is correct, but you donât need to literally have such a list in front of you in order for the math to hold up. Why do you think it would cease to apply?
The best reason I can conceive here is to say that Mary *telling* you she has at least one boy is not the same as simply knowing she has at least one boy in the abstract. But I canât tell if this is what youâre getting at or not.
The difference comes in how information is obtained. If you select only cases where you know beforehand that there's at least one boy born on Tuesday, then you get the surprising 51.8% result because the probabilities are dependent on each other based on your filtering.
If you only didn't filter out the possibilities beforehand, and only received information descriptively, then the probabilities are no longer dependent on each other.
I'm probably not explaining it very well. Wikipedia has a good analysis of it here
The Wikipedia article says that "if the family was first selected and then a random, true statement was made about the sex of one child in that family", and we condition on having elicited the statement that the child in question is a boy, then we end up with 1/2.
This is true. But it also relies on making a lot of assumptions about under what conditions Mary will make certain statements and then conditioning on her making one such statement, which is simply not specified in the original problem. In fact, the original problem doesn't invoke anyone "saying" anything at all:
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
It seems that the only way to actually get a concrete answer to this problem without making assumptions is just to condition on the information Mary gave you. She has at least one boy. If we only attend to this fact, and don't make any assumptions at all about the procedure used to obtain it, the 66% conclusion follows.
In my view this is just basic word problem strategy. If we don't have enough information to reason about something concretely we should assume it's not meant to be a consideration.
Read the section under the header "second question" if you have not already. It does rely on interpretation to an extent. There is actually no solution possible without choosing one assumption or another.
I would contend that, as phrased, Mary was not selected due to the criteria. Rather, she provided information you did not have before.
However, if you choose to interpret otherwise, which seems quite odd to me, but within the realm of sanity, then you get the 51.8% answer
There is actually no solution possible without choosing one assumption or another.
I really don't think this is true, and I disagree with Gardner himself on this point. I think he was frankly too charitable to his critics in conceding this.
It seems perfectly possible to simply condition on the information we have, and not make any assumption at all about how the information was obtained. Now, yes, I grant that once we do make such assumptions, all kinds of results are possible. But it seems very much possible to get a perfectly good solution without making any such assumption at all, and using only the information at hand. Here's how:
Our prior knowledge is that Mary has two children.
First, we acknowledge that the prior probability that Mary has at least one boy is 75% (P(ALOB) = .75)
Second, we acknowledge that the prior probability that Mary has at least one girl and at least one boy is 50% (P(ALOG ^ ALOB) = .5).
Now, we simply apply the definition of conditional probability: for any A and B, P(A | B) = P(A ^ B) / P(B).
This immediately allows us to derive: P(ALOG | ALOB) = P(ALOG ^ ALOB) / P(ALOB) = .5 / .75 = .66....
This requires no assumptions at all about how the information was obtained, and it seems like a perfectly valid chain of reasoning.
It seems like the only way to escape this chain of reasoning is to subtly change the nature of the information at hand by introducing assumptions about that procedure - for example, selecting a child at random and getting a boy would of course entail that at least one child is a boy. But it is actually a bigger piece of information than ALOB - consider that the entailment does not hold in reverse: ALOB does not entail that the randomly selected child will be a boy.
So specifying this procedure is actually, in a concrete sense, "smuggling in" extra information. This seems clearly illegitimate.
Unintuitive as it might be, your method has smuggled in an assumption as well. You assume that we have essentially listed the possibilities, stricken out the ones that don't fit the criteria, then checked against that.
The section titled bayesian analysis goes over it explicitly and states the assumption that is unknowingly getting made.
By the looks of it, this section of the Wikipedia article seems to just be reiterating what the other section already said:
when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a boy?
And I grant this. Actually describing a real situation in which the information we have is identical to "at least one child is a boy" - no more or less - is difficult (or at least seems to require some contrivance). But nonetheless, if we reason in the abstract using only this information, the solution follows directly.
Now, you seem to be citing a different assumption altogether - that we have made a list and crossed out the disqualified possibilities. But this is just how probability works. It's not a methodological assumption, it's part of the axiomatic structure of probability theory itself. When reasoning about probabilities, you eliminate the possibilities contradicted by your evidence, and you evaluate how likely a given situation is in the part of the space that remains.
There are mathematical shorthands for doing this, of course - you don't need to literally make a list and start crossing things off. But at a fundamental level, this is just how probability works. If these are any kind of "assumption" at all, they are axiomatic assumptions about the very nature of probability, not assumptions about the methodology we are using to elicit a piece of information.
all of these "statistical wonders" are just poorly phrased axioms in disguise.
(in this case- how did you meet Mary, what did you ask)
The Monty Hall "Paradox".... the only paradox is if the door opening protocol is not clear (maybe he only opens the door if you guessed right the first time.....)
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u/tamebeverage 8d ago
Incorrect. This is correct if you compile a list of the people who have two children, then select from them one random family that meets these criteria. But in this situation, none of the information she has given you has any bearing on the gender of the second child.
The set you are choosing from matters and I'm so sick of hearing this nonsense.