I think there's an annoying thing where by saying "hey, here's this neat problem, what's the answer" I've made you much more likely to actually get the answer!

What I really wanted to do was transfer the experience of writing a simulation for a related problem, observing this result, assuming I had a bug in my code, and then being delighted when I did the math. But unfortunately I don't know how to transfer that experience over the internet :(

(to be clear, I'm totally happy you wrote out the probabilities and got it right! Just expressing something I was thinking about back when I wrote this blog)