Thread: Monty Hall goes Car-Goat-Key

Ann and Art face three doors marked 1, 2, 3. Behind the doors are placed,
randomly, a car, a goat, and a key. The couple wins the car if Art finds
the car, then Ann finds the key.

First, Art (with Ann removed from the scene) opens a door; if car is not
behind it, he opens a second door. If he fails to find the car, they lose.

If he does find the car, then all doors are closed and Ann gets to open a
door; if key is not behind it, she is allowed to open a second door.
If she fails to find the key, they lose.

Ann and Art do not communicate except to make a plan beforehand.

Using the best strategy, what is the probability that they win the car?

Ok; I'll give you a hint: the probability (with best strategy) is 2/3.
Now this puzzle becomes: HOW do they do it?

There are six cases in which the car, key and goat can be randomly organized.
(C car, G goat, K key)

C G K
C K G
K G C
K C G
G C K
G K C

The strategy could be they would use is that Art would choose Doors 1 and 2, while Ann will choose Doors 3 and 2. why?

2/3 cases have C on the left or in the middle.
2/3 cases have K on the right or in the middle

1/2 cases have C on the left or in the middle AND K on the right or the middle... hmm.

No Macsters, but a bit of the right idea.

Remember that total probability is 2/3.
Since it it 2/3 for Art to find car, then Ann must be able to always
get Key at first door or 2nd door she opens.

They need a plan that makes that possible.

Bump de bump...

No takers?

Well, guess time's up!

The plan:
Bob picks 1:
if it's a goat, he picks 2 next
if it's a key, he picks 3 next

Alice picks 3 ;
if it's a goat, she picks 2 next
if it's a car, she picks 1 next

There are 6 combo's:
C G K
C K G
G C K
G K C
K C G
K G C

These 2 combo's are impossible:
K C G [1]
G K C [2]
These 2 combo's mean SUCCESS:
C G K
G C K
Leaves 2 combo's:
C K G [3]
K G C [4]

[3]: since [1] is impossible, then Alice knows
if she gets G that C K G is only possibility.
So she opens door2.

[4]: since [2] is impossible, then Alice knows
if she gets C that K G C is only possibility.
So she opens door1.