Earlier quoted context omitted.
The probability that he would make that statement given what his children are is a different question than the probability that the other child is also a boy given that he made that statement. Your conditional probabilities are correct, but your conclusion about what they mean for the original question is flawed. N = Total number of ways to have 2 children over 7 days = 14^2 = 196 B = Two sons born on Tuesday. O = Ex…
The problem I have is that you're assuming that each of the 27 outcomes has equal probability, but the chance that we received the information in the way we did makes that a flawed assumption. The best analogous problem is the German Tank Problem: http://en.wikipedia.org/wiki/German_tank_problem If we have destroyed a single German tank with a serial number 100, we can at least to begin to make an estimate on the siz…
I already showed the probability that we would receive the message the way we did, assuming we're sampling fathers with two children, and its pretty low. If we drop the sampling assumption, it would go even lower. But that's irrelevant to the actual question, because we've already won that lottery. I've also shown the probability that we would get the statement we got given that the father had at least one son born on a Tuesday, but again, we already won that lottery.
If you still insist, can you please stop talking in hand-wavy fake math and show some actual concrete numbers? To start, if each of the 27 possibilities are not equally likely, what are the actual probabilities and why?