I don't have that many family members, so I don't know who all those people are. Anyway I'm just gonna pick at random because I am too old and too tired for this shit.

Uh, deal? I walk up to the third door and pull a gold ball from it. I shrug at the madman.

Also, it doesn't say I can only open one box, and all three doors should still have at least one gold ball remaining.

Sorry fam, I don’t negotiate with game theorists.

Commenting before reading other comments

I'm so confused by the added complexity here. If there's a gold ball behind every door, the choice of door doesn't matter.

We were asked to pull a ball from behind one of three doors, not figure out what two dudes know about balls in a grid.

I choose the following option:

Kick the madman in the nuts, open all the doors and take all of the balls, switch the trolly track myself, save the people, sell the balls, then go back and tie the madman to the tracks and ask him to pick the door with the black and blue balls. If he doesn't choose his own fly as the door, TROLLEY.

I think the ball is at C3.

Albert knows that Bernard doesn't know where it is exactly, that means the ball can't be in the first two rows, because if it was that would mean there's a chance for the ball to be in column 5 or 6, in which case Bernard would known right away where it is exactly (as those columns have only one ball). This means the ball must be in row C or D and column 1, 2, 3 or 4. We know Bernard was able to deduce the exact position of the ball from this information, which means it can't be column 1 (as there are 2 balls still remaining there). That means it must be in column 2, 3 or 4. We know Albert was able to deduce the exact position if the ball from this, and since row D has 2 balls still in play (collumn 2 and 4), meaning Albert wouldn't be able to guess which one it was, that means it's in row C, as row C has only one ball left in play.

Now idk what the correct choice would be statistically. If I remember correctly, the Monty Hall problem states that there's 2/3 of a chance you'll get it correctly if you switch doors, but since a gold ball was pulled out from behind the first door (the ball at C3 is gold), I think that means there's a 2/3 chance of the first door (the one from which the ball was pulled out of) being the one with all gold balls (since it contains 2/3 of all gold balls). In that case I think the probabilities cancel out? Which means it doesn't increase your chances whether you switch or not.

I don't see no question.

Hoping to select the door with a goat, that will headbutt the switch for me.

C3. I used to do these logic puzzles at work.

Both Albert and Bernard know the ball is at a certain row and at a certain column, respectively. Albert first admits two things:

1. He doesn't know where the ball is. Aka, the row containing the ball has more than one ball in it. This isn't important now, but will be re-used when he asserts it again.
2. Bernard doesn't know either. If Bernard knew where the ball was simply from the column, it'd be because that column only had one ball in it. Since he knows that Bernard doesn't know given just the row, each ball in that row is in a column that contains more than one ball.

This eliminates rows A and B, since B5 and A6 are the only balls in their columns. For this to work, Bernard now has to understand the above.

Then Bernard admits that now he knows where the ball is. Since we can eliminate or ignore each ball in A and B, it can either be balls D2, C3, and D4 and not any ball in column 1 using logic from #2.

At this point, Albert knows whether the ball is in C or D. If the madman told him the ball was in row C, then he would instantly know the ball is C3 given the elimination of C1 and D1. If he was told it was row D, then he still wouldn't know.

However, Albert admits the former, which tells us it's C3.

Can I try again until I get a silver ball? Please.

Smullyan would be proud.

Sounds to me like you're allowed to see the color of the ball before you take it out. This means it doesn't matter which door you open, a gold ball will be available to pull and you win. Checkmate.