Tuesday, March 9, 2010

The Monty Hall Problem Solved

      Monty Hall was the emcee of the game show “Let’s Make A Deal.” The contestant had a choice of three doors to open; behind one was a prize, and goats were hidden behind the other two. After the contestant chose one of the doors, Hall, who knew where the prize was, opened one of the doors behind which was a goat. The contestant was then given a choice: he could either open the door which was his original choice, or he could switch and open the other closed door.
      Not surprisingly, most people opted to stay with their original choice. It seems logical that since there are now only two doors from which to choose, there is a 50/50 chance the prize is behind either door.
      Unfortunately, that’s wrong! The chances of winning are twice as good if one switches! In fact, this problem has been so often explored by mathematicians that it has been formally named the Monty Hall problem. For those not interested in the mathematics, I will defer the explanation to the end of this blog.
      But recently I came across an interesting story. According to a March 4th article on MSNBC, scientists tested six pigeons with an apparatus with three keys. The keys lit up white to show a prize was available. After the birds pecked a key, one of the keys the bird did not choose deactivated, showing it was a wrong choice, and the other two lit up green. The pigeons were rewarded with bird feed if they made the right choice.
      In the experiments, the birds quickly reached the best strategy for the Monty Hall problem — going from switching roughly 36 percent of the time on day one to some 96 percent of the time on day 30.
      On the other hand, 12 undergraduate student volunteers failed to adopt the best strategy with a similar apparatus, even after 200 trials of practice each.
      Humans tend to figure out the probable outcomes ahead of time, whereas the pigeons just allowed their experience to guide them. For this particular problem, their method of learning was better suited to the problem.
      In order to ascertain that the mathematics was correct, I ran the whole scenario on my computer, using random numbers for all the variables. The following table shows the results after two replications of 100 trials each.


Description
1st. Replication2nd ReplicationCombined

Change
Winners3278%3162%6369%

Losers922%1938%2831%

Original
Winners2237%1530%3734%

Losers3763%3570%7266%
For the trials in which the original decision was changed, 69% were winners, versus only 34% winners in the trials where the decision was made to stick with the original door.
      Now for the mathematics: To see why the apparently illogical choice of switching is actually better, one must understand that before the host opened one of the three doors, the contestant did not know the location of the prize, and thus when he or she chose a door, the contestant had a 1-in-3 chance of being right. That does not change even after the host opened a door. If the probability of the first door the contestant chose remained the same, and there were only two doors left, that meant the remaining unopened door must have had a 2-in-3 chance of being right — that is, it had twice the chance of holding the prize.
      It seems complicated, but the experience of the pigeons proves that it works.

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