This solution was derived independently
but the result is equivalent to
the answer already
given by Mark Tilford
and the confident tone of
a suggestion by Greg Martin
led to the realization that this does provide
the highest possible probability of winning the car.
Alice and Bob can win with a probability of 2⁄3,
or 4 times out of the 6 possible hidden-item arrangements.
The present strategy actually guarantees
that Bob will surely find the Key if Alice finds the Car.
This surprising result cannot be improved because
there is no way to reduce the 1⁄3 probability
that the Car remains behind Alice’s unchosen third door.
Imagining a repeating cycle of doors,
. . . Left - Middle - Right - Left - Middle - Right - Left -
. . . ,
the hidden items may have two orderings:
1⁄2 chance that the items are ordered as:
. . . Car - Goat - Key -
Car - Goat - Key -
Car . . .
1⁄2 chance that the items are ordered as:
. . . Car - Key - Goat - Car - Key - Goat -
Car . . .
The basis of this solution is that
Alice and Bob assume that the items are in order # 1.
If # 1 is indeed the order then
Alice and Bob will both find their desired items because
each will correctly assume which door to open second
if their initial choice is wrong.
(Which doors are chosen first doesn’t even matter.)
The trick is to also have some chance of winning if
the items are in order # 2.
In this case Alice and Bob can win only if both are lucky enough to
choose their first doors correctly
as they will make incorrect adjustments for their second doors.
Therefore:
Bob’s first choice should be one door to the right (cyclically) of
Alice’s first choice
because the Key is one door to the right of the Car
in order # 2.
This strategy can be specified as:
Alice first chooses the Middle door and is content
if it reveals the Car being sought.
If it reveals the Goat, however, Alice then chooses the Left door;
had the Key been revealed, Alice chooses the Right door second.
Bob first chooses the Right door and is content
if it reveals the Key being sought.
If it reveals the Car, however, Bob then chooses the Middle door;
had the Goat been revealed, Bob chooses the Left door second
because that is the next door
cyclically to the right of the Right door.
Here is how this works out for the 6 possible item arrangements.
Ordering # 1: ... C - G - K - C - G - K - C - G - K - C - G - K - C ...
(3 arrangements, Left Mid Right Left Mid Right Left Mid Right (Left)
all winners) C G K G K C K C G (K)
------------- ------------- -------------
Alice 1st choice G K C!
good 2nd choice C! C! (moot)
Bob 1st choice K! C G
good 2nd choice (moot) K! K! (K)
Ordering # 2: ... C - K - G - C - K - G - C - K - G - C - K - G - C ...
(3 arrangements, Left Mid Right Left Mid Right Left Mid Right
1 winner) C K G K G C G C K
------------- ------------- -------------
Alice 1st choice K G C!
wrong 2nd choice G K (moot)
Bob 1st choice G C K!
wrong 2nd choice C G (moot)
SIDE NOTE. A different strategy is almost as successful
while allowing Alice to enjoy
making at least one choice randomly.
Alice chooses the first door randomly.
If it is the Middle door, Alice doubles the fun by
choosing the second door randomly as well;
otherwise Alice next chooses the Middle door.
One of Alice’s choices is sure to be the Middle door.
Bob robotically chooses the Left and Right doors.
The Middle door will not be chosen by both contestants.
As the Car and Key are behind different doors,
the prospect of winning improves on pure randomness by eliminating a
possible same-door (Middle & Middle) combination
of the contestants’ choices
and thus increasing the likelihood of a different-door combination.
Of the 12 possible outcomes, 1⁄2 are winners:
Car is in Middle Goat is in Middle Key is in Middle
(4 outcomes, (4 outcomes, (4 outcomes,
all winners) 2 winners) no winners)
Left Mid Right Left Mid Right Left Mid Right
G C K C G K C K G
------------- ------------- -------------
Alice chooses L and M G C! C! G C K
Bob chooses L and R G K! C K! C G
Alice chooses M and R C! K G K K G
Bob chooses L and R G K! C K C G
Left Mid Right Left Mid Right Left Mid Right
K C G K G C G K C
------------- ------------- -------------
Alice chooses L and M K C! K G G K
Bob chooses L and R K! G K C G C
Alice chooses M and R C! G G C! K C
Bob chooses L and R K! G K! C G C