Monty Hall

No doubt most readers are familiar with the Monty Hall problem, so bear with me while I repeat it. The reason is that I was reading an article the other day about how such a simple exercise in probability was so misunderstood by so many people. The problem is this:

Suppose you are on a game show, and you are given the choice of three doors: Behind one door is a car, behind the others, goats. You pick a door, say No. 1 and the host, who knows what's behind the other doors, opens another door, say No. 3, to reveal a goat. He then asks you if you want to switch your choice from door No. 1 to door No. 2. Is it to your advantage to switch?

Apparently when this question was posed in an American magazine in 1990, and the columnist gave the correct answer, she received 10,000 letters, and her readers "acted as if she'd advocated ceding California back to Mexico".

Surveys show that 13% of Americans don't believe that plants create oxygen, 24% don't believe that light travels faster than sound, and 35% believe that you can make milk radioactive by boiling it, but on the Monty Hall question some 92% believed the published answer was wrong.

Apparently close to 1,000 Ph.D.s wrote in, many of them mathematics professors, who seemed especially irate. One professor wrote "How many irate mathematicians are needed to change your mind?" Someone from the US Army Research Institute remarked "If all those Ph.D.s are wrong the country would be in serious trouble."

But the answer was correct.

One leading mathematician of the 20th century, Paul Erdos, said "That's impossible" and then when presented with a formal mathematical proof of the correct answer, he still didn't believe it "and grew angry." Only after a colleague arranged for a computer simulation with hundreds of trials did Erdos concede that he was wrong.

Part two on probability decisions later.