Can you explain the essence of the Monty Hall paradox and why it seems counterintuitive?

Monty Hall paradox is a classic probability problem that seems counterintuitive due to perception of probabilities.

The essence of the paradox:

• You have three doors: behind one is a prize, behind two are empty.
• You choose one door.
• The host, who knows what is behind the doors, opens one of the remaining doors that does not have a prize.
• Then he offers you to switch your choice to the remaining closed door.

Intuitively, it seems that the chances are 50-50, but in reality, the probability of winning if you switch is 2/3, and if you stay with the initial choice, it is 1/3.

This contradiction arises because the host provides additional information by opening a door without a prize, changing the probability distribution. By switching, you effectively choose between two doors instead of one.