It seems that, with two doors closed, everything comes down to fifty-fifty. However, the odds are not equal: the way the presenter chooses which door to open changes everything.
The Monty Hall Problem
Riddle statement
In a contest there are three doors. Behind one there is a car and behind the other two there are goats. You choose a door. The presenter, who knows where the car is, always opens one of the other two doors that a goat has, never opens your door and always offers you to change to the only door that is closed.
If you decide to change doors, does your probability of winning the car improve? What is that probability?
Show solution
Solution
Answer: Yes, switching improves the probability of winning. By switching, you win with probability $2/3$.
At first, your door has probability $1/3$ of hiding the car. The other two doors, together, have probability $2/3$.
When the presenter opens a goat, he is not eliminating just any door. He knows where the car is and always avoids opening it. Therefore, the probability of $2/3$ that was distributed between the two unchosen doors does not disappear: it is concentrated in the only unchosen door that is still closed.
Like this:
if you keep your initial choice, you win with probability $1/3$;
if you change, you win with probability $2/3$.
Another way of looking at it: switching wins exactly when your first choice was a goat. Since at the beginning there were two goats and a car, this happens in two out of three cases.
Therefore, it is advisable to change the door.