I feel like the Monty Hall problem is counter-intuitive the first time you see it.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The answer is yes, you should ALWAYS switch your choice. The reasoning is as follows: at the beginning you have a 1 in 3 chance of selecting the correct door. After the host shows you another door, there are only 2 doors to select from there. You initially chose the first door which had a probability of $1/3$ of being the correct door. Now, because all the probabilities within a set of choices must always add to $1$, we can conclude that the 2nd door is correct with a probability of $2/3$. So indeed, switching your guess is to your advantage.

I feel like the Monty Hall problem is counter-intuitive the first time you see it.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door (say No. 1). Then the host, who knows what's behind all the doors, opens another door (say No. 3) that is guaranteed to have a goat. He then says to you, "Do you want to now pick door No. 2?" Is it to your advantage to switch your choice?

The answer is yes, you should ALWAYS switch your choice. The reasoning is as follows: at the beginning you have a $1$ in $3$ chance of selecting the correct door. After the host shows you another door, there are only $2$ doors to select from there. You initially chose the first door which had a probability of $1/3$ of being the correct door. Now, because all the probabilities within a set of choices must always add to $1$, we can conclude that the 2nd door is correct with a probability of $2/3$. So indeed, switching your guess is to your advantage.