Conditioning by throwing runs away

P(A | B) means: among the occasions when B happens, how often does A happen too? A simulation can do that literally. Run the experiment, throw away every run where B is false, and count A among the survivors.

From the slides: Probability and what happens when you learn something and Why should anyone care about chance?.

What to notice

Conditioning shrinks the sample space: the struck-out runs in each log are the ones that could not have happened, given the information.
The two pieces of news about the coins sound alike but keep different runs. “The first toss was heads” keeps two of the four outcomes; “at least one toss was heads” keeps three. That is why the answers are 1/2 and 1/3.
In Bertrand's box, the gold–gold box is twice as likely as the gold–silver box to hand you a gold coin, so seeing gold is evidence for it: 2/3, not 1/2.