User talk:Elen of the Roads/Montypeluciano dAbruzzu

The Solution

Vos Savant's solution only works if - like a typical game show host - Monty Hall knows what's behind the doors only reveals the prize at the end. To do this, he must make sure never to open the door with the car behind it.

If Monty pick's his door at random, he will pick the car 1/3 of the time, and you will pick the car 1/3 of the time. There is no advantage to switching

You pick	Monty picks	The other door has the	result if switching	result if staying
Car	Goat	Goat	Goat	Car
Goat	Goat	Car	Car	Goat
Goat	Car	Goat	Goat	Goat

However, if Monty knows what's behind the doors, and is only going to reveal the prize at the end, he has to pick a door with a goat. This gives quite a different result. In this case, if you have picked a goat, Monty has to pick the other goat and leave the car hidden. You still only have a 1/3 chance of picking the car to start with, but if you switch, 2/3 of the time (the two times you picked the goat) the other door will have the car behind it.

You pick	Monty picks	The other door has the	result if switching	result if staying
Car	Goat	Goat	Goat	Car
Goat	Goat	Car	Car	Goat
Goat	Goat	Car	Car	Goat

This is true even if you use the more detailed mathematical technique known as conditional probability, which takes into account that if you pick the car, Monty has two goats to choose from.

Case 1: Car behind door 1	Case 2: Car behind door 2	Case 3: Car behind door 3
Player picks door 1	Player picks door 1	Player picks door 1
No matter what the host does, switching away from the car loses	Host must open door 3; switching away from the goat wins	Host must open door 2; switching away from the goat wins
One case where switching loses	Two cases where switching wins

Other versions of the original problem

Other approaches to explaining the problem

The solution using mathematical notation

Variations on the problem

Door 1	Door 2	Door 3	result if switching	result if staying
Your door has the car	Monty opens to show a goat	has a goat	goat	car
	has a goat	Monty opens to show a goat
Your door has a goat	Monty opens to show a goat	has the car	car	goat
Your door has a goat	has the car	Monty opens to show a goat	Car	Goat

Door 1	Monty opens door 2	Door 3	result if switching	result if staying
Car	Goat	Goat	Goat	Car
Goat	Car	Goat	Goat	Goat
Goat	Goat	Car	Car	Goat

Door 1	Door 2	Monty opens door 3	result if switching	result if staying
Car	Goat	Goat	Goat	Car
Goat	Goat	Car	Goat	Goat
Goat	Goat	Car	Car	Goat

If Monty doesn't know where the car is, there is no advantage to switching.