>>13423980If you pull out a random ball in the set where silver balls are present, that's a 1/2 chance of getting a gold ball.
The odds of switching to door #2 are 2/3 that it's 100% gold balls. So there is a 2/3 chance door #1 has some silver balls behind it.
For the last part, Albert knows Bernard doesn't know, so it can't be B (or else Bernard could hear 5 and know where it is). By the same logic it can't be A (or else Bernard could hear 6). So Albert heard C or D and Bernard knows this now. Bernard now knows the answer, so it can't be 1 or else he still wouldn't know if it's C1 or D1. So now Bernard knows which of 2-3-4 it is. Since Albert knows as well, it has to be C3 (if it was D, he still wouldn't know between D2 and D4).
So the ball that was thrown is gold. If there are all gold balls behind door 1 that's still a 100% chance. If there are silver balls, then the madman pulled out one gold ball, and now you must pull out a second. The question is - do you know which box he pulled the ball out of? Say you shake all 3 and hear which one has 1 ball in it.
If you know which box has only one ball, the odds of taking the second ball from the same box is 2/3. So if you choose Door 1, that's 1/3 chance of having all gold + 2/3 chance of having silver * 2/3 chance of guessing the second gold ball = 7/9. If you choose Door 2, that's 2/3 chance of having all gold + 1/3 chance of having silver * 1/2 chance of pulling out a gold ball with no prior information = 5/6. The second fraction is bigger (83% vs 78%) so you go with Door 2.
If you don't know which box has only one ball, your odds of pulling out gold on Door 1 go down, and I'm too lazy to figure out the last bit of math to prove it but Door 2 has better odds anyway so you should always switch to Door 2 and pick a ball.
QED, pick Door 2 and pray to Jesus that you're in the correct timeline (the probability of success, conditioned on your soul being in a state of grace, is 1)
