[Math] Family Has Two Children

probability

I know this has been answered a dozen times, and I know entirely how to get the 2 possible answers. My question revolves around the phrasing of this question, aren't (1) and (2) the exact same question?

A neighbor of yours has two children. Assuming that the gender of a child is like a coin flip, it is most likely that the neighbor has one boy and one girl, with probability of a half. The other possibilities are a quarter each (either two boys or two girls).

(1) Suppose that you ask the neighbor whether she has any boys, and she said yes. What is the probability that one child is a girl?

(2) Instead, suppose that you happened to see one of her children passing by, and it was a boy. What is the probability that the other child is a girl?

In both questions, it states that we have knowledge of the neighbor having a boy. The first question, she says yes (she has at least one boy). The second, we see at least one boy. Both asks the probability of the other child being a girl. Clearly, as I've read the other questions on this topic, as well as the wiki this is a play on words. So I guess I'm having trouble depicting which one is 2/3 and which is independent of each other and thus 1/2.

Best Answer

This problem demonstrates the difference between facts and events. And to understand it, you need to be painfully exact with procedure.

  1. Identify every independent outcome in your sample space, even those that may contradict what you will learn.
  2. Assign a probability to each outcome.
  3. An "event" is a set of outcomes defined by some common property. In these questions, the first event you need is the set of outcomes where you obtain the knowledge that there is at least one boy. It isn't necessarily the set where the fact is true, it is the set where you obtain the knowledge by the procedures described.
  4. The second event is the subset of the first where there is also a girl.
  5. The answer is found by adding up the probabilities for the outcomes in each set, and dividing the sum for the subset in #4 by the sum for the set in #3.

In Question #1, there are four outcomes: BB, BG, GB, and GG. Each has a probability of 1/4. The mother will answer "yes" for the set {BB, BG, GB}, and "no" otherwise. The subset {BG, GB} also has a girl. So the answer is (1/4+1/4)/(1/4+1/4+1/4)=2/3.

In Question #2, there is another factor needed to define the outcomes. Which child did you see? So there are eight outcomes: BB1, BB2, BG1, BG2, GB1, GB2, GG1, and GG2; where the number indicates which child you see. The probability of each is 1/8.

You saw a boy, meaning the set for #3 is {BB1, BB2, BG1, GB2}. Notice how some cases, where the fact "there is a boy" is true, that are not included. The subset where there is also a girl is {BG1, GB2}. The answer is {1/8+1/8)/(1/8+1/8+1/8+1/8)=1/2.

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