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…

B_{T}G_{M } B_{T}G_{T }B_{T}G_{W} B_{T}G_{Th }B_{T}G_{F }B_{T}G_{S} B_{T}G_{Sn}

G_{M}B_{T } G_{T}B_{T }G_{W}B_{T} G_{Th}B_{T }G_{F}B_{T }G_{S}B_{T} G_{Sn}B_{T}

**B _{T}B_{M } B_{T}B_{T }B_{T}B_{W} B_{T}B_{Th }B_{T}B_{F }B_{T}B_{S} B_{T}B_{Sn}**

**B _{T}B_{M } B_{T}B_{W} B_{T}B_{Th }B_{T}B_{F }B_{T}B_{S} B_{T}B_{Sn}**

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?