Michael E. Brandt, Ph.D., is an assistant professor in the Department of Psychiatry and Behavioral Sciences at the University of Texas Medical School. His research interests are in the analysis of brain biomedical images and signals. He has been programming in C since 1983. Most of his applications are in signal and statistical data analysis. Dr. Bandt can be contacted at the University of Texas Medical School, Dept. of Psychiatry, P.O. Box 20708, Houston, TX 77225
Marilyn vos Savant is listed in the Guinness Book of World Records Hall of Fame for "Highest IQ." She is the author of a question-and-answer column in the Sunday edition of Parade Magazine. What began as a question posed to Ms. vos Savant by a reader ended up becoming a major controversy. In one of her columns she posed the following puzzle: a contestant is on a game show and has a choice of three doors. Behind one door is a brand new automobile, behind the other two are goats. The contestant picks a door, say, Number One, and the host, who knows what's behind each door, opens another one, say, Number Three, behind which is a goat. He then gives the contestant the option of picking door Number Two. Should the contestant switch choices?
Marilyn asserted that the contestant should switch since the first door has a one-third chance of winning, but that door Number Two has a two-thirds chance.
Marilyn subsequently received thousands of letters concerning her response. Overall, 92% of the readers from the general public completely disagreed with her, and of the letters from universities, 65% were against her answer. The latter included letters from several well-respected Ph.D.'s who accused Ms. vos Savant of intellectual irresponsibility.
In a follow-up Parade column, Ms. vos Savant clearly explained her answer and proposed a test to math classes all across the country. Her challenge was to set up a probability experiment in which one student plays contestant and another plays host. There are three paper cups labeled No. 1, No. 2, and No. 3. With the contestant's eyes closed the host throws a die until a one, two or three are rolled, then hides a penny under the corresponding cup. The contestant then throws a die the same way and selects a cup. The host lifts up one of the two unchosen cups that he knows is a loser. The contestant "stays" by uncovering the chosen cup to see if it reveals a penny. This sequence is repeated 200 times keeping track of the correct number of wins. The game is then played an additional 200 times by testing the switching strategy. Following the last instruction the contestant switches by selecting the cup not chosen by anyone for the presence of a penny.
To test Ms. vos Savant's answer to this problem I decided to write a simulation in C which you can use to decide for yourself. Therefore I have chosen not to provide her explanation to the problem in this article, except indirectly by discussing the workings of the program.
The Program
Any C programmer worth his or her salt would take the above experimental specification as a personal challenge to write a Monte Carlo simulation. Listing 1 is my personal response to such a challenge. You can conceptualize the problem by using a three-bit binary number to represent the state of the doors in response to the actions taken by the contestant and host. You can then use the bitwise operations, for which C is particularly adept, to change the state of this three-bit number. Bit 0 represents the zeroth door, bit 1 represents the first door, and bit 2 the second door. The simulation is designed to play N games of "not switching" followed by an equal number of "switching" trials.The simulator begins by generating a random number between zero and two, equivalent to the host placing the prize behind a door (variable called car). Then, the status of the three-bit number (doors) is updated. The state of the doors containing goats is represented as a variable called noprize:
~doors & 7Next, a random number is produced representing the contestant's selection (choice), followed by the host revealing the first door:
noprize & ((~(1 << choice)) & 7)This latter door number is represented as open. For the not switching scenario, all trials in which choice matches car are tallied and the percentage of these to the total number of trials (N) is printed out. Notice that in the no-switch condition, a match is not dependent in any way on the variable open.For switching, the contestant creates a new choice which is the NOT of the old choice ANDed with the NOT of open. That is,
not_choice = ~(1 << choice) & 7 not_open = ~(1 << open) & 7and
choice = not_open & not_choice.All trials in which this new choice matches doors are tallied and the percentage of these to the total number of trials (N) is also printed out. Notice in this case that a match is dependent on open, and therein lies the key to the problem!Figure 1 shows the output from several Monte-Carlo trials using the program in Listing 1 for increasing numbers of trials. As can be clearly seen, the probability of "winning" when not switching doors approaches one-third, while the probability of winning after switching approaches two-thirds as asserted by Ms. vos Savant.
Discussion
In a recently published book entitled Innumeracy, John Allen Paulos describes a human condition (analogous to illiteracy) in which there is a fundamental lack of proficiency in basic mathematics as well as the principles of probability and statistics. Although the problem described in this article is not as simple as it seems at first blush, the fact that 65% of university personnel (including many highly trained Ph.D.s) were equivocal in their analysis demonstrates the seriousness of this problem. It also indicates that the computer is still not regarded as a tool for testing the correctness of certain claims, or for simulating reality. That represents a particularly virulent form of innumeracy known as computer illiteracy.
Reference
Paulos, John Allen. 1990. Innumeracy, Mathematical Illiteracy and its Consequences. New York: Vintage Books.