Alice picks a card from a deck of cards. Bob guesses what her card is. Say Bob guesses the King of Hearts. Alice says her card is a heart, and gives Bob the option to keep his current guess or to change his guess. Is he more likely to guess correctly if he changes his guess?
I think this is Monty with 52 doors. When Alice reveals that her card is a heart, she is opening 39 doors (corresponding to the other 3 suits) to reveal goats. Bob is therefore more likely to win by changing his guess, since the probability he guessed correctly at the outset is $\dfrac{1}{52}$ so the probability one of the other 12 hearts is Alice's card is $\dfrac{51}{52}$. Each of these 12 cards is equally likely to be Alice's card, so the probability Bob wins if he switches his guess to another heart is $\dfrac{1}{12}\cdot\dfrac{51}{52}$.
Is this correct? Or does Monty Hall not apply here? If it does not apply, why does it not apply?
I think Monty Hall dose not apply here, since there is a major difference between your question and Monty Hall: Monty Hall's choice will not release any information about whether the car is behind the door of your current choice (since he cannot open the door you have chosen), so when Monty Hall opens the door, it increases the probability of other doors that he do not open but remains the probability of your choice unchanged. However, in your question, once Alice say her card is a heart (she is opening 39 doors), the probability of any heart cards (including Bob's current choice) increase.
In a word, Monty Hall's choice dose not release information about your current choice, whlie Alice's words dose release information about whether Bob's current choice is right.