Comment
Brian Hayes makes use of the "Lets Make a Deal" puzzle to make the point that computer simulation is a good way to sort out controversies such as the one Marilyn vos Savant created by asserting there is an advantage to switching one's initial choice of three doors, one of which has a prize behind it, once one door has been opened that does not reveal the prize. Actually, he inadvertently makes a quite different point which is that Monte Carlo simulations are only as good as the structure of the model used to run them. He gets a surprise free outcome by structuring the probabilities a la vos Savant as 1/3 for the original choice and 2/3 for switching to the other remaining door. Run that case 100,000 times, and yes, you'll get the vos Savant answer that it makes sense to switch choices from the original door. That doesn't make it the correct answer.
The initial statement of the problem as three doors with a prize randomly placed behind them does have a 1/3 probability of being the right choice prior to the opening of a door that doesn't contain a prize. Once door is opened without the prize, the problem changes to two doors each of which is equally likely to have the prize behind it. There is no advantage to switching doors. If a second door is opened without a prize behind it, one doesn't need a computer simulation to know the probability of it being behind the final door.
Monte Carlo simulations are most useful when no explicit analytical structure will give a precise answer or there is doubt about data accuracy. Usually there are many random variables that have different frequency distributions that are developed empirically. Some of the variables are correlated with one another, others are entirely independent. In these situations the model generates a distribution of outcomes which can actually be of more predictive value than a single exact number based on static assumptions.
Bruce Sampsell
Chapel Hill, NC 27514
posted by Robert Sampsell
July 15, 2008