Originally Posted by Barkis is Willin'
Right, and if the daughter, whose gender you have revealed....
I haven't revealed the gender of any child. All I've told you is that they're not both boys. This is a subtle distinction, but it's at the heart of the problem.
Try this experiment on for size: Take two quarters and flip them. Write down the number of heads. After you repeat this about 100 times, look at the ratio of times you wrote "one" to times you wrote "two". It will be pretty damn close to 2 with very high probability.
If you don't want to do it by hand, here's the R code for an equivalent experiment:
M <- matrix(runif(2 * 100) < 0.5, ncol = 2)
N <- apply(M, 1, sum)
ratio <- length(N[N == 1]) / length(N[N == 2])
I ran this a thousand times, and the average ratio was 2.07. Only 10% of the ratios were less than 1.5. If the true value were one, that basically wouldn't happen.