During my Shadowcon talk, I invited participants to think about their fundamental reactions to problems, make their own conjectures, before seeking to confirm or refute them. I apologize for the delay, but here I include the solutions, and hope that you got a chance to play around with the questions before reading.
If your friend has two children, and one of them is a boy, what are the odds the second one is a boy as well?
Intuitively it “feels right” to react with, irrespective of what gender the first child is, the second child has a 50/50 chance of being a boy because these are independent events. But as my suspicious pause suggested, the extra knowledge about the first child does change up the odds a bit. Since my friend has two children, there are 4 total possible outcomes.
Knowing that one child is a boy, cuts the number of possible outcomes down to 3. Of these three, only in one of them is the second child a boy as well. Therefore the probability of the second child being a boy, given we know that one of the children was a boy, is 1/3 (not ½).
Imagine now that we have some additional information: the boy was born on a Tuesday. Does it change the probability of the second child being a boy from 1/3?
As long as we know it’s a boy, why would the day he was born affect the odds of the second child?
Writing out all the possible pairings where one of the children is a boy and Tuesday-born…
BTGM BTGT BTGW BTGTh BTGF BTGS BTGSn
GMBT GTBT GWBT GThBT GFBT GSBT GSnBT
BTBM BTBT BTBW BTBTh BTBF BTBS BTBSn
BTBM BTBW BTBTh BTBF BTBS BTBSn
There are a total of 27 outcomes, out of which 13 pairs have two boys, with one of them born on a Tuesday. That extra information bumped us up about 15% from 1/3 to 13/27!
What felt helpful about the opportunity to acknowledge and honor your intuitive response to the question before seeking its validity?