So I’m familiar with the Monty Hall problem and the boy/girl paradox (and by familiar, I mean I know the right answers). Now I’m trying to devise a subtly deceptive version of my own so as to torment my friends who refuse, despite my insistent recommendations, to read the straight dope. Of course, I have to be sure I’m right before I do that. Here goes nothing:
You’re at the office. The front desk calls and tells you that two people are waiting for you. As you walk down the hall and turn the corner, you see that one of them is a woman, but you haven’t seen the other person yet. What is the probability that the other visitor is a man (ignoring all sociological factors, etc–assume that the genders might as well be sides of a coin)?
Correct answer: 1/2.
Now, you’re at the office. The front desk calls and tells you that two people are waiting for you. As you walk down the hall, just before you turn the corner, you hear what is unmistakably a woman’s voice (don’t be sexist–it’s not the receptionist, and there are no other people in the room aside from the receptionist and the two visitors). What is the probability that the other visitor is a man?
Correct answer: 2/3.
By hearing a woman, you know that at least one of the visitors is a woman. But by seeing a woman, you know WHICH of the visitors is a woman, which makes all the difference.
In both cases you know one is a woman. Whether by sight our sound why does it matter?
Your question doesn’t seem to care though which person is a man. Just the odds that one might be and in both cases you have determined with certainty that one is a woman leaving you a 1:2 chance the other is a male.
In both cases, the visitors are differentiatable. In the first case, you have “the visitor you see” and “the visitor you don’t see;” in the second case, you have “the visitor you hear” and “the visitor you don’t hear.” In both cases, you know that a specific visitor is female.
This is similar to variations of the boy/girl paradox in that the correct answer shifts based on the children being differentiated in any fashion: oldest/youngest, tallest/shortest, first through the door/second through the door, or the one you see/the one you don’t.
There’s no difference between the one you see/the one you don’t and the one you hear/the one you don’t.
Sad to say, I can’t remember ever receiving a visitor (a business visitor, anyway) at the office who was female. Dammit - law enforcement support and IT Security were the two worst professions I could have chosen.
Does the female sound cute?
That said, I would say it largely depends upon the industry. In industries such as those I mentioned above, females are a somewhat rare article (the law enforcement organization I worked for was 5% female; the IT organization was probably about the same, despite being “one of the best places for single mothers, women, and GLBTs) to work” according to the much lauded industry magazines.
At the risk of sounding like a male pig (which, trust me, I’m not - except during “frisky time”), the industry in question is important. I would suspect that there are a lot of women in traditionally feminine industries, compared to some of the ones I’ve worked in.
Ahh, but when those 25 year old accountants come to do audits… well, we stare. Like a bunch of male pigs.
I agree. In each case you know that one specified person – the one you can see, or the one you can hear – is a woman. That gives you no information about the other person. There’s a 50/50 chance that the unseen/unheard person is a woman.
It’s still 1/2 because you asked what the probability is that the other visitor is a man. The only two cases are woman-woman or woman-man. There’s no case where it’s man-woman, because I’m listing the heard visitor first. Here, there is no chance that the visitor you heard is a man.
ETA: Instead, work in that one visitor is standing at the desk and another is sitting in a chair. Then you hear the voice but you don’t know which visitor spoke. Then ask what the probability that the sitting person is man. It’s 1/3 since you know that M-M is impossible.
Yeah, I have no intention of starting one of these arguments for the millionth time, but I really would have sworn up and down that OP has it right here… Scenario one we will encounter one person of which we have no information, clearly the probability of male is 1/2. Scenario two we will encounter two persons, and our only information is that at least one of them will be female. Possible combinations for this encounter is FM, MF and FF. Probability of male must be 2/3.
So suppose you’re blind, and you walk into a room to meet two visitors. One visitor says, “Hi, my name is Mary. What’s the probability that my colleague is a man?”
This is exactly the same scenario as put forth in the OP. Do you think the probability that the colleague is male is 2/3? If so, why would a specific person be more likely to be male just because he/she is in a room with Mary?
chessic sense: you are onto something. my wording could be clearer. however, when i say “the other visitor” you still don’t which visitor is the OTHER visitor, because you don’t know which one you’ve heard. again, my wording could have been better, but it’s still technically correct.
the point of using the hearing v. seeing distinction is to create two situations, one in which the two visitors are differentiated and one in which they are not. based on the answers of the first few posters, i think i’ve succeeded in creating a subtle deception, with chessic’s caveat in mind.
i haven’t actually differentiated them by saying that i hear one and not the other, because again you don’t know which.
Disagree. You’ve differentiated them by designating one as the one you hear, and one as the one you don’t, much like you first scenario differentiated them by designating one as the one you see, and one as the one you don’t.
But, is it fair to assume that all three of those combinations are equally likely?
Let’s look at it like this: Before we have any information at all, there are 8 possibilities, all equally likely:
Person A is Male, Person B is Male, Person A speaks.
Person A is Male, Person B is Female, Person A speaks.
Person A is Female, Person B is Male, Person A speaks.
Person A is Female, Person B is Female, Person A speaks.
Person A is Male, Person B is Male, Person B speaks.
Person A is Male, Person B is Female, Person B speaks.
Person A is Female, Person B is Male, Person B speaks.
Person A is Female, Person B is Female, Person B speaks.
When we hear a female voice, we can rule out #1, #2, #5, and #7 (in which the person who speaks is male), leaving us with #3, #4, #6, and #8.
Out of those 4 possibilities, how many are there in which the other person (the person who does not speak) is male? Two of them: #3 and #6. So the probability that the other person is male is 2/4, or 1/2.
No, you’ve missed the point. When you say, as you do in #8 for example: Person A is female, person b is female, person b speaks, you’ve PRESUMED to know which person speaks. The correct breakdown would be this:
A woman speaks, so it can’t be MM.
A woman speaks, so it could be MF. (If it helps, think of them as standing in this arrangement, i.e. the man on the left and the woman to the right.)
A woman speaks, so it could be FM. (Woman is on the other side of him now.)
A woman speaks, so it could be FF.
Compare this to sight. Assume that as you look into the room, you sweep from left to right.
You see a woman, so it can’t be MM.
2 .You see a woman, so it can’t be MF. (Because you’re looking left to right, and would see a man first.)
Thudlow, another way to think of it is this: if, as in your numbers 3 and 6, you hear a woman, then the other person is necessarily a man, and the two are differentiated. however, in 4 and 8, hearing one woman doesn’t tell you that you HAVEN’T heard the other, because you don’t know what either of them sounds like. obviously you haven’t heard them both, but you can’t know whether it’s person A or B–it’s an equally likely possibility. for this reason, your numbers 4 and 8 collapse into a single possibility: Person A is female, Person B is female, a female spoke. You’re left with not 2/4 then, but 2/3.
There’s a hidden assumption that you’ve made, let’s make that assumption plain by slightly altering the scenario:
Two turn up to see you at reception at the same time, both have a 50-50 chance of being male or female. The receptionist cannot decide which one to send up to you first so she (or indeed he:D) flips a coin to decide which one to send up first. The person she sends up is female, what is the probabilty that the one waiting at reception is also female?
Whenever in doubt, just write out Bayes’ rule explicitly. In the two scenarios, I’ll assume that the person you see first is chosen randomly (perhaps they are running in circles around the room, and you interrupt them to see a random person) and that the person you hear first is chosen randomly (perhaps they are talking back and forth, and you get within earshot while a random person is talking.)
The answer is 0.5 in both scenarios. The probability that both are women given that you’ve seen a women is:
Pasta, no. You’re again assuming that whatever word you substitute still constitutes a differentiation. It doesn’t. By seeing one woman you know which one you’ve seen, but by hearing a woman you don’t. Again, if you’re looking from left to right, then you make the following error:
P(seen w|mw)P(mw) = 0.5. The probability that you see a woman first, given that the man is first in your line of sight, is 0. Therefore the sum is 1.5, making your ratio 1/3 which supports my conclusion of 2/3.
No, again because it’s not differentiated. Think the original boy girl paradox, something like: A family has two children. One is a boy. What is the probability that the other is a girl?
B B
B G
G B
2/3. You don’t weight the first one twice just because it could be either boy. That’s accounted for in the other outcomes. Look that one up if you’re not convinced. Wikipedia has a good explanation.
