Last I checked, your edits haven’t changed which answer is correct in your scenario. As you’ve explained, the Ace is impossible given your set-up.
(By the way, I thought that the earliest version of your wording was perfectly adequate, provided that the reader was accustomed to puzzles given in a “propositional” form. Otherwise, I expect, the reader will naturally assume something like the “algorithmic” scenario that I’ve been describing.)
In my scenario, the information given is not about which propositions are true about the outcome, but rather about which algorithms are controlling the outcome.
To highlight the difference, let me flesh out my story.
Let K be the set of card-hands that contain at least one King, let A be the set of card-hands that contain at least one Ace, and let Q be the set of card-hands that contain at least one Queen.
I’m programming the card-dealing robot. I’ve prepared two different algorithms, either of which could be used by the robot:
Algorithm 1: Choose a hand uniformly at random from K ∪ A, and then deal that hand.
Algorithm 2: Choose a hand uniformly at random from Q ∪ A, and then deal that hand.
These are two different algorithms. If the robot is programmed with one of them, it cannot be programmed with the other. That is, the algorithms are mutually exclusive. Moreover, I am going to use one or the other of them. These two algorithms exhaust all of the possibilities.
In other words, of the two algorithm-descriptions above, exactly one of them will truthfully describe the robot’s actual algorithm.
I flip a coin to determine which algorithm will control the robot. After the coin flip, I program the robot accordingly, supply it with cards, and bring you to the table with the robot.
You know all of the above.
Now the robot deals you a hand, face down. Based on what you know, which is more probable: that the hand contains a King, or that the hand contains an Ace?
Thanks for this. I understand your point now.
I was misreading this:
In my scenario, the information given is not about which propositions are true about the outcome, but rather about which algorithms are controlling the outcome.
Last I checked, your edits haven’t changed which answer is correct in your scenario. As you’ve explained, the Ace is impossible given your set-up.
(By the way, I thought that the earliest version of your wording was perfectly adequate, provided that the reader was accustomed to puzzles given in a “propositional” form. Otherwise, I expect, the reader will naturally assume something like the “algorithmic” scenario that I’ve been describing.)
In my scenario, the information given is not about which propositions are true about the outcome, but rather about which algorithms are controlling the outcome.
To highlight the difference, let me flesh out my story.
Let K be the set of card-hands that contain at least one King, let A be the set of card-hands that contain at least one Ace, and let Q be the set of card-hands that contain at least one Queen.
I’m programming the card-dealing robot. I’ve prepared two different algorithms, either of which could be used by the robot:
Algorithm 1: Choose a hand uniformly at random from K ∪ A, and then deal that hand.
Algorithm 2: Choose a hand uniformly at random from Q ∪ A, and then deal that hand.
These are two different algorithms. If the robot is programmed with one of them, it cannot be programmed with the other. That is, the algorithms are mutually exclusive. Moreover, I am going to use one or the other of them. These two algorithms exhaust all of the possibilities.
In other words, of the two algorithm-descriptions above, exactly one of them will truthfully describe the robot’s actual algorithm.
I flip a coin to determine which algorithm will control the robot. After the coin flip, I program the robot accordingly, supply it with cards, and bring you to the table with the robot.
You know all of the above.
Now the robot deals you a hand, face down. Based on what you know, which is more probable: that the hand contains a King, or that the hand contains an Ace?
Thanks for this. I understand your point now. I was misreading this: