Example 7.23: the Monty Hall problem

The Monty Hall problem illustrates a simple setting where intuition often leads to a solution different from formal reasoning. The situation is based on the game show Let's Make a Deal. First, Monty puts a prize behind one of three doors. Then the player chooses a door. Next, (without moving the prize) Monty opens an unselected door, revealing that the prize is not behind it. The player may then switch to the other non-selected door. Should the player switch?

Many people see that there are now two doors to choose between, and feel that since Monty can always open a non-prize door, there's still equal probability for each door. If that were the case, the player might as well keep the original door.

This approach is so attractive that when Marilyn Vos Savant asserted that the player should switch (in her Parade magazine column), there were reportedly 10,000 letters asserting she was wrong. There was also an article published in The American Statistician. As reported on the Wikipedia page, out of 228 subjects in one study, only 13% chose to switch (Granberg and Brown, Personality and Social Psychology Bulletin 21(7): 711-729). In her book, The Power of Logical Thinking, vos Savant quotes cognitive psychologist Massimo Piattelli-Palmarini as saying "... no other statistical puzzle comes so close to fooling all the people all the time" and "that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer."

One correct intuitive route is to observe that Monty's door is fixed. The probability that the player has the right door is 1/3 before Monty opens the non-prize door, and remains 1/3 after that door is open. This means that the probability the prize is behind one of the other doors is 2/3, both before and after Monty opens the non-prize door. After Monty opens the non-prize door, the player gets a 2/3 chance of winning by switching to the remaining door. If the player wants to win, they should switch doors.

The excellent Wikipedia entry referenced above provides additional intuitive tools, as well as variants and history.

One way to prove to yourself that switch is best is through simulation. In fact, even deciding how to code the problem may be enough to convince yourself to switch.

SAS
In SAS, we assign the prize to a door, then make an initial guess. If the guess was right, Monty can open either door. We'll switch to the other door. Rather than have Monty choose a door, we'll choose one, under the assumption that Monty opened the other one. We do this with a do until loop (section 1.11.1). If our initial guess was wrong, Monty will open the only remaining non-prize door, and when we switch we'll be choosing the prize door.

data mh;
do i = 1 to 100000;
  prize = rand("TABLE",.333,.333);
* Monty puts the prize behind a random door;
  initial_guess = rand("TABLE",.333,.333);
* We make a random initial guess;
* if the initial guess is right, Monte can 
                  open either of the others;
* which means that player can switch to either 
                  of the other doors;
  if initial_guess eq prize then do;
    new_guess = initial_guess;
    * choose a door until it's different from
                  the initial guess-- that's
                  the door we switch to;
    do until (new_guess ne initial_guess);
      new_guess = rand("TABLE",.333,.333);
   end;
    end;
* If the initial guess was wrong, Monte can rule 
          out only one of the other doors;
* which means that we must switch to the right door;
  if initial_guess ne prize then new_guess = prize;
  output;
  end;
run;

To see what happened, we'll summarize the resulting data set. If our initial guess was right, we'll win by keeping it. If switching leads to a win, the new guess is where the prize is. We create new variables to indicate winning using a logical test (as in section 1.4.9).

data mh2;
set mh;
win_by_keep = (initial_guess eq prize);
win_by_switch = (new_guess eq prize);
run;

proc means data = mh2 mean;
var win_by_keep win_by_switch;
run;

The MEANS Procedure

Variable                 Mean
-----------------------------
win_by_keep         0.3332700
win_by_switch       0.6667300
-----------------------------

Note that these two values sum to exactly one. If we chose the prize initially, we always win by keeping it, and if we did not choose the prize initially, we always win by switching. In any event, the simulation supports the conclusion that we should switch.



R

In R, we write two functions. In one, Monty opens a door, choosing at random among the non-chosen doors if the initial choice was correct, or choosing the one non-selected non-prize door if the initial choice was wrong. The other function returns the door chosen by swapping. We use the sample() function (section 1.5.2) to randomly pick one value. We then use these functions on each trial with the apply() statement (section B.5.3).

numsim = 100000
doors = 1:3
opendoor = function(x) { 
 if (x[1]==x[2])
    return(sample(doors[-c(x[1])], 1)) 
 else return(doors[-c(x[1],x[2])])
}
swapdoor = function(x) { return(doors[-c(x[1], x[2])]) }
winner = sample(doors, numsim, replace=TRUE)
choice = sample(doors, numsim, replace=TRUE)
open = apply(cbind(winner, choice), 1, opendoor)
newchoice = apply(cbind(open, choice), 1, swapdoor)

To display the results, we use the cat() statement to show how text and variables can be integrated.

> cat("without switching, won ",
+ round(sum(winner==choice)/numsim*100,1)," 
+ percent of the time.\n", sep="")
without switching, won 33.2 percent of the time.
> cat("with switching, won ",
+ round(sum(winner==newchoice)/numsim*100,1)," 
+ percent of the time.\n", sep="")
with switching, won 66.8 percent of the time.