The answer is not, "of course YES!" The correct answer is, "I don't know, can you be more specific?"
The only reason why you think it is correct, is because Marliyn vos Savant said so. Her original answer to the question (although the question was widely know before her) appeared in Parade magazine on September 9, 1990. she wrote that the "correct" answer to this question was to switch doors, because switching doors gave you a higher probability of winning the car (2/3 instead of 1/3). She got lots of responses from Mathematics PhDs and other intelligent people that said she was wrong (although many of them were incorrect too).
Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? — Craig F. Whitaker Columbia, Maryland
I have bolded the important part of this logic question. What is ambiguous in that statement is:
Does Monty Hall always open a door? (What would it be to your advantage to switch doors if he only opened a losing door when you picked a winning door? Answer: No)
Does Monty Hall always open a losing door? (The question specifies that he knows where the car is, and this particular time he showed a goat behind one. What would your chances be if he randomly opened a door? i.e. The Monty Fall question or what if sometimes he chooses to show winning doors.)
Does Monty Hall always open a door you did not pick?
The basics of this logic puzzle have been repeated more than once, and many times they aren't specified well enough to give the "correct" answer of 2/3.
A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male? — Stephen I. Geller, Pasadena, California
Did the fellow look at both dogs before responding "Yes," or did he pick up a random dog and discovered it was a male and then responded "Yes."
Say that a woman and a man (who are unrelated) each has two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy. Can you explain why the chances that the woman has two boys do not equal the chances that the man has two boys? My algebra teacher insists that the probability is greater that the man has two boys, but I think the chances may be the same. What do you think?
How do we know that the women has at least one boy? Did we look over the fence one day, and see one of them? (Answer: 50%, same as man)
The question has even tripped up our very own Jeff Atwood. He posed this question:
Let's say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl. What are the odds that person has a boy and a girl?
Jeff goes on to argue that it was a simple question, asked in simple language and brushes aside the objections of some that say that the question is incorrectly worded if you want the answer to be 2/3.
More importantly though, is why the woman volunteered the information. If she was speaking the way normal people do, when some one says "one of them is a girl," inevitably the other is a boy. If we are to assume this is a logic question, with the intent of tripping us up, we should ask that the question is more clearly defined. Did the woman volunteer the sex of one of her children, randomly selected, or is she talking about the set of her two children.
It is clear that the question is poorly worded, but people don't realize it. When similar questions are asked, where the odds are much greater to switch, people either realize that it must be a trick (and question the motive of the host), or get the "correct" answer of switching as in the one hundred doors question. This is further supported by the fact that doctors when asked about the likelihood of a woman having a particular disease after testing positive (they need to determine if she has the disease, or it is a false positive), they are better at arriving at the correct answer, depending upon how the question is phrased. There is a wonderful TED Talk that half way through covers this very case.
He described the probabilities associated with a breast cancer test: 1% of women tested have the disease, and the test is 90 percent accurate, with a 9% false positive rate. With all that information, what do you tell a woman who tests positive about the likelihood they have the disease?
If it helps, here’s the same question phrased another way:
100 out of 10,000 women at age forty who participate in routine screening have breast cancer. 90 of every 100 women with breast cancer will get a positive mammography. 891 out of 9,900 women without breast cancer will also get a positive mammography. If 10,000 women in this age group undergo a routine screening, about what percentage of women with positive mammographies will actually have breast cancer?