Coding Horror: Finishing The Game

If they tell you "we have two kids", you ask them if at least one of them is a girl and they answer "yes", then indeed the probability of the other kid beeing a boy is 2/3.

If they tell you "we have two kids, and one of them is a girl", then by all rules of rational discourse the probability for the other kid being a boy is 1.

Things start to get interesting if the talking member of the couple is a mathematician (so the rules of ordinary discourse do not apply) and he says "We have two kids, and at least one of them is a girl". Now you might say that this is equivalent two the first case. But you could also go bayesian and look at the possible conversations with couples containing at least one mathematician that have two kids. Say you have 100 couples. On average, 25 of these will have two boys and tell you "We have two kids, and at least one of them is a boy", another 25 will have two girls and tell you "We have two kids, and at least one of them is a girl". The remaining 50 couples will have one boy and one girl. Assuming no gender bias on behalf of the speaking mathematician (ha!), 25 of these will tell you "We have two kids, and at least one of them is a boy", while the remaining 25 will say "We have two kids, and at least one of them is a girl." So out of the 50 couples that tell you "We have two kids, and at least one of them is a girl," 25 will have two girls, and 25 will have a boy and a girl. So in this case, the correct answer is 1/2.

(This is essentially the argument from the intro on this page that was linked from the discussion on coding horror: http://www.overcomingbias.com/2008/10/my-bayesian-enl.html )

Here, I'll prove it for you:

    ;; Have a child, with an equal probability of the child being a boy or a
    ;; girl.
    (defun make-child ()
      (if (eql (random 2) 0)
          'boy
          'girl))

    ;; Make a family of n members, with each child having an equal probability
    ;; of being either a boy or a girl.
    (defun make-family (n)
      (do ((family nil)
           (i 0 (+ i 1)))
          ((>= i n) family)
        (push (make-child) family)))

    ;; As in the story, meet someone at a party who informs us that (1) she or
    ;; he has two children, and (2) that at least one of these children is a
    ;; girl.  E.g.,
    ;;
    ;; Me: "Hi there, what's your name?"
    ;; Cute lady: "Jennifer"
    ;; Me: "Nice to meet you, Jennifer, I'm Mark.  What brings you here?"
    ;; Jennifer: "Well my husband is out of town on a business trip, so I
    ;;            wanted to come here and catch up with some old friends of
    ;;            mine.  Fortunately I was able to get a baby sitter for my
    ;;            two kids on such short notice.  One of the kids, Meg, has
    ;;            to be up early in the morning for dance practice and..."
    ;; Me: "Husband?  Damn, all the good ones are taken."
    ;; Jennifer: "What?"
    ;; Me: "Nothing.  Hey, hang on while I work out the probability that your
    ;;      other child is a boy, based on the information that you've already
    ;;      given me."
    ;; Jennifer: "You're weird.  Have a good evening."
    ;;
    ;; This function works by calling (make-family 2) repeatedly until we get a
    ;; family that meets both of these criteria, then returns a representation
    ;; of said family.  This is a precise analogy for the story: just as the
    ;; fictional person at this party, here we take the space of all possible
    ;; conceptual representations of a family of two, then discard any of these
    ;; that *does not* include at least one female child.  THIS IS THE ONLY
    ;; REASONABLE WAY TO HANDLE THE INFORMATION THAT JENNIFER RELATES IN THE
    ;; ABOVE EXCHANGE.
    (defun meet-at-party ()
      (do ((family (make-family 2) (make-family 2)))
          ((find 'girl family) family)))

    ;; Perform the (meet-at-party) scenario num-total times, and then return
    ;; the fraction of those times in which the family contained a boy.
    (defun run-test (num-total)
      (let ((num-with-boys 0))
        (dotimes (i num-total (float (/ num-with-boys num-total)))
          (if (find 'boy (meet-at-party)) (incf num-with-boys)))))