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Ask Marilyn: Still Wondering About the Game Show Problem?
Kitt Carlton-Wippern of Thousand Oaks, California, writes:
Marilyn: Look, this isn't so difficult a problem, but people need to think in terms of sets and not just elements. Three doors, you chose one, there are now two sets. Set A is the set you own. From the original draw it has a probability of 1/3 of holding the prize and encompasses one element (in this case, a selected door). Set B is the set you don't own. From the original draw it has a probability of 2/3 of holding the prize and encompasses two elements.
One of the two elements in Set B is not the prize; the game show host shows you that element (the one that is not the prize) in Set B, removes it from Set B, sets it aside; Set B now has one element. But from the original draw, Set A still has probability 1/3 of holding the prize (this is the key, no change from the original draw or selection), and Set B by definition must still have probability 2/3 of holding the prize irrespective of the number of elements it now contains (which is now down to one element).
Again, key point: no change from the original draw/selection. Why? Because there have been no changes made to the original conditions nor circumstances under which the selection, when selected, was made. The probability remains 1/3 you hold the winning set consisting of one element. So 2/3 must therefore remain the probability of the other set (modified after the original selection process, yes, but the modification was shown to not affect whether or not the prize was still in play). Nothing in any way has changed that alters the probability that the set you have, the set you selected--Set A--is any different from 1/3 (that you hold the winning set).
And so, with responses like the ones you received: "Your answer to the question is in error [switch sets]. But if it is any consolation, many of my academic colleagues have also been stumped by this problem," is it any wonder that the NASA manned space program has been cancelled? Could this be particularly due to the fact America's scientists, engineers and mathematicians clearly don't know about real probability theory (as is clearly evidenced by this example, and specifically, probability theory when applied to reliability and failure analysis, as was uncovered by Richard Feynman, Nobel laureate in physics, member of the Rogers Commission during the Challenger shuttle disaster; and resurfaced by Admiral Harold Gehman [Columbia Accident Investigation Board]? Of even greater concern is the response, "You [Marilyn] made a mistake, but look at the positive side. If all those Ph.D.s were wrong, the country would be in some very serious trouble," sent by a member of the U.S. Army Research Institute.
Well, I have some really bad news for everyone. All those Ph.D.s are wrong. The country is in serious trouble, and maybe the paradigm of home schooling ought to be applied to a college education as well. People routinely complain about the poor education in grades K through 12. The above is clear and compelling evidence that mediocrity in education is apparently not limited to primary and secondary school but continues through all levels of college well into the post-doctoral programs.
Thank you, Kitt. I think!