Question: A market research field worker needs to interview married couples
A market research field worker needs to interview married couples about use of a certain product. The researcher arrives at a residential building with three apartments. From the names on the mailboxes downstairs, the interviewer infers that a married couple lives in one apartment, two men live in another, and two women live in the third apartment. The researcher goes upstairs and finds that there are no names or numbers on the three doors, so that it is impossible to tell in which of the three apartments the married couple lives. The researcher chooses a door at random and knocks. A woman answers the door. Having seen a woman at the door, what now is the probability of having reached the married couple? Make the (possibly unrealistic) assumptions that if the two men's apartment was reached, a woman cannot answer the door; if the two women's apartment was reached, then only a woman can answer; and that if the married couple was reached, then the probability of a woman at the door is 1/2. Also assume a 1/3 prior probability of reaching the married couple. Are you surprised by the numerical answer you obtained?
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