Unlike the original scenario, your visitors now remain undifferentiated, and there’s a 2/3 chance you’ll find a man in the room.
In your hearing example, you can distinguish the visitor you heard from the visitor you did not hear. Therefore, the answer is 1/2. When I say nothing else, I mean nothing else.
I have a theory about why the classic “2/3” answer to the boy-girl puzzle is so counter-intuitive, and it’s all in the wording. The statement is normally worded something like “I have two children. One is a boy. What is the probability that the other is a girl?”
Now, as Chronos says, “the other” would normally be enough to indicate a particular individual, which as we have discussed here would make the probability 1/2. But look at that statement again: “One is a boy” is short for “at least one is a boy”, which itself is can be expanded to “either one is a boy or both are boys”. But if you state it like that, referring to “the other” makes no sense, because you have already referred to both of them. Who is this “other”?
IF you assume
(1) that visitors are EQUALLY likely to be male as female (probably a very unlikely assumption depending on reason for visit), AND
(2) that a male-female (or female-male) pair is exactly twice as likely as female-female (probably unlikely even if (1) is true), AND
(3) that males and females speak equally often, AND
(4) that the chance of hearing two voices rather than one does NOT depend on gender mix,
THEN 1/2 is the correct answer.
It is not pedantic to insist that these assumptions be recognized. Probability is the craft of reasoning from one’s imperfect knowledge; to ignore that knowledge (or pretend it to be different from reality) is to miss the whole point of a probability problem.
I think it depends whether the receptionist means “I saw both visitors, and at least one of them is a woman”, or she means “I saw one of the visitors, and she was a woman”. Again, it comes down to whether the visitor is identified in any way. If so, it eliminates one of the MF/FM possibilities, depending on which arbitrary criteria you use to order them. And that just leaves you with FF and FM (or MF), so the probability is 50%.
Even with reasonable assumptions of the type septimus lists, the problem is often confused by the wording, as Ximenean mentioned. For example, “One is a boy.” could be referring to a specific child. That’s why I like ** freewillzyx**'s sexist pig" scenario; the wording at least is clear.
Hmm, I just realised that I blithely assumed that the receptionist was herself female…
Yours, a sexist.
Exactly right, which makes the wording of the boy-girl problem with “the other” really, really terrible.
Just about the only reasonable way to say it is “So-and-so has two children, of which at least one is a boy. Given just this information, what is the probability that both children are boys?”. (Well, you could dress it up as Chronos did as well, though that amounts to the same thing; the final question in that case was still “What is the probability that both children are boys?”).
There’s one other way I can think of to semi-naturally put it: King so-and-so had two children. Upon King so-and-so’s death, one of his children became king, upon whose death the other child ascended to the throne. What is the probability that this other child was a girl?
The answer, if you decode the wording the way you’re intended to and keep in mind the convention of male-preference primogeniture, would be 2/3. That the first successor was a male amounts to exactly the same thing as there being at least one male out of the two successors, regardless of relative ages; “the other child” is well-defined as the second successor, who would be either the unique girl or the younger boy. As noted above and in the classical boy-girl problem, the three scenarios “Elder son, younger son”, “Elder son, younger daughter”, and “Elder daughter, younger son” would then be equally likely, with the second successor being a girl in 2 out of these 3.
(Of course, it’s worth noting that, with this kind of setup, we also have, even more strongly biased: King so-and-so had two children, his two successors the throne. What is the probability that the second successor was a girl? Answer: 3/4)
It can’t be assumed that hearing one allows you to distinguish the speaker from the other. You don’t know the position of the visitors, and those positions aren’t necessarily static. In other words, then, it’s not like saying that “the eldest” is a woman, which is an exclusive differentiation. I agree with previous comments that the whole thing more or less hinges on the word “other”…when you say “other” people think you have designated a certain one, when really you have only designated either one. Think if I had thrown in: “One of the visitors is wearing a red shirt, and the other isn’t. Just before turning the corner you hear a woman’s voice.” By hearing that, you haven’t distinguished the visitors from one another on the basis of speech any more than on the basis of their clothing. However, you can now apply Bayes’ rule knowing from the speech that it is not a man.
As others explained very clearly, the reason why the answer is 1/2 comes down to the application of Bayes’ rule. The fact that we hear a woman, if we assume that whoever speaks was determined at random, gives us important information about the “original” probability of the visitors’ genders. I had been treating the two as independent because I thought I could state that the speaker was female as a given, which was clearly wrong.
Phrasing things in terms of distinguishability or not may be confusing or misleading. The semantic quibbles about what it means to distinguish things are a red herring. It’s really a question of information. The problem with the hearing example can be illustrated like this: You presumably know more relevant information in that case than just the fact “At least one of the two people here is a woman”.
That is, if the only information you were conditioning on was “At least one of the two people here is a woman”, then the probability that there is one man and one woman would of course be 2/3. But whenever you condition on further information, probabilities can shift, often drastically. In particular, if we also introduce the random variable of the gender whose voice you hear, and then condition on the further information “Furthermore, the gender of the voice you hear is a woman”, then the probability that there is one man and one woman shifts back down to 1/2.
You want the setup to have only the random variable “Are there two men, two women, or one of each?” and for the listener to have only the information “There is at least one woman” to condition on, but the way you’ve described it, the natural interpretation is that there is both more relevant variation in the space of possibilities (“Also, which gender will be heard?”) and more relevant information available to the listener (“The voice heard will be female”). You could ignore this if the extra variation was independent from the variation you are interested in, but, of course, it is not (Females are more likely to be heard if there are more females around).
Bayes’ rule illustrates this well, but it’s worth understanding at an intuitive level as well what is happening here. You’ve introduced more random variables than you wanted to, and allowed more information to leak through than you wanted to, shifting probabilities.
Actually, reading your last paragraph, you seem to have a pretty good grasp on that. But I just wanted to say, it has nothing fundamentally to do with issues of “distinguishability”, except to the extent that you can phrase such issues in terms of what information one does and does not have about what random variables correlated with the random variable one is interested in. The children in the monarchical succession example I gave before are quite “distinguishable”, yet the answer is still 2/3 there.