Answers to Monty Hall revisited
As promised, the answers to Monty Hall revisited. They can be found behind the cut.
Obviously, there is no strategy such that you can guarantee a win. This would mean that the you'd always be able to find the car. Since there's one door unopened, the probability of you finding the car is 2/3.
In the same way, the probability of your friend finding the car keys is also 2/3. So if we'd use the strategy "Both you and your friend pick two doors at random", the chances of winning are 4/9, or a little under 50%.
Of course, we want to do better than that.
First, what if both you and your friend pick the same two doors. Then it doesn't matter which of the two doors has the car or the key, as long as you open both of them. There are three ways in which we can choose two doors (namely: {1,2}, {1,3}, and {2,3}), and only one of these three will be correct. The chance of winning is 1/3. That's worse than opening doors randomly.
Next try. You open {1,2}, and your friend opens {1,3}. To determine the probability of winning, we split it up in what is behind door 1:
1/3: goat. The car and the keys are behind doors 2 and 3, but you don't know in which order. Since there are two possible orders, the chance of choosing the right order is 1/2
1/3: car. Congrats, you have done your thing. However, the keys are either behind door 2 or 3. Your friend has a chance of 1/2 of picking the right one.
1/3: keys. Same as earlier, but now you have a chance of 1/2 of finding the car.
Total chance of winning = 1/3*1/2 + 1/3*1/2 + 1/3*1/2 = 1/2. That's better than guessing at random!
Still, we can do better. So far, the choice of the second door was completely independent from the result of the first door. This means that the order of opening doesn't matter. Now what if we let the choice of the second door depend on the result of the first door. Here's a strategy with the best possible chance:
Your strategy:
Open door 1.
If you see a goat: open door 3.
If you see the keys: open door 2.
If you see the car: open either, you've done your thing.
Your friend's strategy:
Open door 2.
If you see a goat: open door 3.
If you see the car: open door 1.
If you see the keys: open either, you've done your thing.
To see what's going on, let's look at what happens to your friend if this strategy leads you to the car. There are three possibilities here:
1) The car is behind the first door. This means that behind the second door are either the keys, or the goat. If there are keys, congrats. If it's the goat, your friend opens the third door which must then have the keys.
2) There's keys behind the first door, and the car behind the second. This means that your friend sees a car, and thus opens door 1, and finds the keys you saw. Congrats, you win.
3) There's a goat behind the first door, and the car behind the third. This means that the keys are behind the second door, so all is right.
As you can see, this strategy is chosen in such a way that whenever you find the car, your friend also finds the keys. The converse is true as well:
There are two cases in which you don't find the car:
1) There's keys behind the first door, and the car behind the third. This means your friend sees the goat, and then opens the third door, and finds the car.
2) There's a goat behind the first door, and the car behind the second. This means your friend sees the car, and then opens the first door, and finds the goat.
So whenever you find the car, your friend finds the keys. And whenever you don't find the car, your friend doesn't find the keys either. Since we already established that the chance of you finding the car is 2/3, the chance of winning the game is also 2/3.
Obviously, there is no strategy such that you can guarantee a win. This would mean that the you'd always be able to find the car. Since there's one door unopened, the probability of you finding the car is 2/3.
In the same way, the probability of your friend finding the car keys is also 2/3. So if we'd use the strategy "Both you and your friend pick two doors at random", the chances of winning are 4/9, or a little under 50%.
Of course, we want to do better than that.
First, what if both you and your friend pick the same two doors. Then it doesn't matter which of the two doors has the car or the key, as long as you open both of them. There are three ways in which we can choose two doors (namely: {1,2}, {1,3}, and {2,3}), and only one of these three will be correct. The chance of winning is 1/3. That's worse than opening doors randomly.
Next try. You open {1,2}, and your friend opens {1,3}. To determine the probability of winning, we split it up in what is behind door 1:
1/3: goat. The car and the keys are behind doors 2 and 3, but you don't know in which order. Since there are two possible orders, the chance of choosing the right order is 1/2
1/3: car. Congrats, you have done your thing. However, the keys are either behind door 2 or 3. Your friend has a chance of 1/2 of picking the right one.
1/3: keys. Same as earlier, but now you have a chance of 1/2 of finding the car.
Total chance of winning = 1/3*1/2 + 1/3*1/2 + 1/3*1/2 = 1/2. That's better than guessing at random!
Still, we can do better. So far, the choice of the second door was completely independent from the result of the first door. This means that the order of opening doesn't matter. Now what if we let the choice of the second door depend on the result of the first door. Here's a strategy with the best possible chance:
Your strategy:
Open door 1.
If you see a goat: open door 3.
If you see the keys: open door 2.
If you see the car: open either, you've done your thing.
Your friend's strategy:
Open door 2.
If you see a goat: open door 3.
If you see the car: open door 1.
If you see the keys: open either, you've done your thing.
To see what's going on, let's look at what happens to your friend if this strategy leads you to the car. There are three possibilities here:
1) The car is behind the first door. This means that behind the second door are either the keys, or the goat. If there are keys, congrats. If it's the goat, your friend opens the third door which must then have the keys.
2) There's keys behind the first door, and the car behind the second. This means that your friend sees a car, and thus opens door 1, and finds the keys you saw. Congrats, you win.
3) There's a goat behind the first door, and the car behind the third. This means that the keys are behind the second door, so all is right.
As you can see, this strategy is chosen in such a way that whenever you find the car, your friend also finds the keys. The converse is true as well:
There are two cases in which you don't find the car:
1) There's keys behind the first door, and the car behind the third. This means your friend sees the goat, and then opens the third door, and finds the car.
2) There's a goat behind the first door, and the car behind the second. This means your friend sees the car, and then opens the first door, and finds the goat.
So whenever you find the car, your friend finds the keys. And whenever you don't find the car, your friend doesn't find the keys either. Since we already established that the chance of you finding the car is 2/3, the chance of winning the game is also 2/3.