On: Think you understand Monty Hall? Try the Tuesday b...

[hwiechers]

I finally understand it.

Here are my events (taking 2 children as given):

  B = I have a boy
  BT = I have a boy and he was born on Tuesday
  2B = I have 2 boys
  B+G = I have a boy and a firl

Using Bayes' Theorem:

  Pr(2B|BT) = Pr(BT|2B)Pr(2B)/c
  Pr(B+G|BT) = Pr(BT|B+G)Pr(B+G)/c
  where c = Pr(2B|BT) + Pr(B+G|BT)

  Pr(BT|2B) = 1/7 + 1/7 - 1/49 = 13/49
  Pr(2B) = 1/4
  Pr(BT|B+G) = 1/7
  PR(B+G) = 1/2

So

  c = 13/49 * 1/4 + 1/7 * 1/2 = 27/49
  Pr(2B|BT and 2C) = (13/49 * 1/4) / (27/49) = 13/27
  Pr(B+G| BT and 2C) = (1/7 * 1/2) / (27/49) = 14/27

They main reason Pr(2B|BT) is around 50% is that Pr(2B|BT) is proportional to Pr(BT|2B) while Pr(B+G|BT) is proportional to Pr(BT|B+G) and Pr(BT|2B) is about 2 * Pr(BT|B+G).

It's much more likely you have a boy born on Tuesday given that you have 2 boys (rather than you have a girl and a boy) so it's more likely that you have 2 boys given that we know you have a boy born on Tuesday.