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.
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