TL;DR

Additional reflections about the Monty Hall problem.

In previous post The Monty Hall problem I took a look at the Monty Hall problem, with a Perl twist.

E7…. user in Twitter was so nice to play a bit with it and had an interesting consideration:

When “ABCPlayer” plays against “RandomMontyHall”, the player wins ~50% of times.

I already observed this phenomenon, and tought that it had to be related to the fact that the ABCPlayer does indeed swap sometimes, so it’s probably equivalent to a random swap for enough runs of the simulation:

package ABCPlayer;
use parent -norequire => 'Player';
sub swaps_with ($self, $unrevealed) {
   for my $alternative ($self->{alternatives}->@*) {
      next if $alternative eq $self->{initial};
      return $alternative eq $unrevealed;
   }
}

But of course at this point I had to brush off my lazyness and do some better analysis.

There are a total of 9 possible random arrangements of the prizes behind the door and player’s initial choices, indicated with round parentheses:

 A   B   C
-----------
(W)  L   L

 W  (L)  L

 W   L  (L)

(L)  W   L

 L  (W)  L

 L   W  (L)

(L)  L   W

 L  (L)  W

 L   L  (W)

The winning prize can be behind door A, B, or C and for each of these three possibilities the initial player’s choice can be, again, door A, B, or C.

The RandomMontyHall host will choose deterministically if the player has a losing door, and randomly otherwise. To account for both possibilities of this random choice, it makes sense to double all these possibilites and indicate the opened door with square brackets:

 A   B   C                A   B   C
-----------              -----------
(W) [L]  L               (W)  L  [L]

 W  (L) [L]               W  (L) [L]

 W  [L] (L)               W  [L] (L)

(L)  W  [L]              (L)  W  [L]

[L] (W)  L                L  (W) [L]

[L]  W  (L)              [L]  W  (L)

(L) [L]  W               (L) [L]  W

[L] (L)  W               [L] (L)  W

[L]  L  (W)               L  [L] (W)

As expected, the opened door always reveals a losing prize. Pairs on the same line are equal, except for the cases where a random choice is done by the host, in which case we show both alternatives.

Now we can apply the ABCPlayer’s tactic to mark which cases yield a swap and which don’t:

 A   B   C                A   B   C
-----------              -----------
(W) [L]  L   keep        (W)  L  [L]  swap

 W  (L) [L]  swap         W  (L) [L]  swap

 W  [L] (L)  swap         W  [L] (L)  swap

(L)  W  [L]  swap        (L)  W  [L]  swap

[L] (W)  L   keep         L  (W) [L]  swap

[L]  W  (L)  keep        [L]  W  (L)  keep

(L) [L]  W   keep        (L) [L]  W   keep

[L] (L)  W   keep        [L] (L)  W   keep

[L]  L  (W)  keep         L  [L] (W)  swap

As expected, there are 9 swaps and 9 keeps. Let’s also add the player’s outcome:

 A   B   C                A   B   C
-----------              -----------
(W) [L]  L   keep W      (W)  L  [L]  swap L

 W  (L) [L]  swap W       W  (L) [L]  swap W

 W  [L] (L)  swap W       W  [L] (L)  swap W

(L)  W  [L]  swap W      (L)  W  [L]  swap W

[L] (W)  L   keep W       L  (W) [L]  swap L

[L]  W  (L)  keep L      [L]  W  (L)  keep L

(L) [L]  W   keep L      (L) [L]  W   keep L

[L] (L)  W   keep L      [L] (L)  W   keep L

[L]  L  (W)  keep W       L  [L] (W)  swap L

Again, as expected there are 9 wins and 9 losses, which also accounts for the ~50% of player’s wins in the long run.

Please stay safe!