Monty Hall Problem Intuition

Solution 1:

I cannot find a flaw in your reasoning.

My own reasoning (if you are interested).

Someone who sticks to his original choice will win if his original choice was correct.

Probability on that: $\frac13$.

Someone who switches will win if his original choice was wrong.

Probability on that: $\frac23$.

Solution 2:

I believe the best way to intuitively understand the Monty Hall problem is by playing the game with a $100$ doors, $99$ goats and one supercar.

I can choose a door, doing so will give me a probability of $1\%$ of choosing the car. The host then opens $98$ doors, showing $98$ goats. At this point I know that the door I chose either contains the supercar (with a probability of $1\%$) or more likely a goat ($99\%$).

Now I'm given the oppurtinity to switch doors, it's clear that doing so will increase my chance of getting the supercar to $99\%$.