The Arguement
Throughout the many years of Let's Make A Deal's popularity, mathematicians have been fascinated with the possibilities presented by the "Three Doors" ... and a mathematical urban legend has developed surrounding "The Monty Hall Problem."
A heated debate began when Marilyn Savant published a puzzle in her Parade Magazine column. One of her readers posed the following question: “Suppose you’re on a game show, and you’re given a 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, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to take the switch?”
Ms. Savant, who’s listed in the Guinness Book of World Records Hall of Fame for “Highest IQ” (228), answered “Yes.” Because of the estimated 10,000 letters she received in response, she published a second article on the subject.
Due to the fervor created by Ms. Savant’s two columns, the New York Times published a large front page article in a 1991 Sunday issue which declared: “Her answer... has been debated in the halls of the C.I.A. and the barracks of fighter pilots in the Persian Gulf. It has been analyzed by mathematicians at M.I.T. and computer programmers at Los Alamos National Laboratory in New Mexico. It has been tested in classes ranging from second grade to graduate level at more than 1,000 schools across the country.”
Most people would expect the chances of winning in both sets of 100 trials to be 1 in 3.
Many people would claim that odds of winning in the second trial is 1 in 2, as only 2 cards are face down
Method
One person put on a bad sportsjacket and dealt three cards face down: 2 jokers, representing the undesirable "zonk" prizes, and the ace of spades, representing a new car.
The second person dressed up as Little BoPeep and chose one of the cards. The dealer then revealed a joker and gave the player an opportunity to choose the other card.
For the first set of 100 trials, the player chose to stick with the original card choice.
For the second set of 100 trials, the player always chose the other face down card. Robert and Karen each took turns and dealer and player.
Player does not change mind after one Joker is revealed
| Trial | Robert | Karen | Karen | Robert |
| 1 | zonk | zonk | zonk | win |
| 2 | zonk | zonk | zonk | zonk |
| 3 | win | win | zonk | zonk |
| 4 | win | win | win | win |
| 5 | zonk | win | zonk | win |
| 6 | zonk | win | zonk | zonk |
| 7 | zonk | zonk | win | zonk |
| 8 | zonk | zonk | win | win |
| 9 | zonk | win | zonk | zonk |
| 10 | zonk | zonk | zonk | zonk |
| 11 | zonk | zonk | zonk | zonk |
| 12 | zonk | zonk | zonk | zonk |
| 13 | zonk | zonk | zonk | win |
| 14 | zonk | zonk | zonk | zonk |
| 15 | win | zonk | zonk | zonk |
| 16 | zonk | zonk | zonk | zonk |
| 17 | win | win | win | zonk |
| 18 | win | win | win | zonk |
| 19 | win | zonk | zonk | zonk |
| 20 | win | zonk | zonk | zonk |
| 21 | win | win | win | zonk |
| 22 | win | zonk | win | zonk |
| 23 | zonk | zonk | zonk | zonk |
| 24 | zonk | zonk | win | zonk |
| 25 | win | zonk | win | win |
| TOTALS | ||||
| win | 10 | 8 | 9 | 6 |
| loose | 15 | 17 | 16 | 19 |
WIN: 33 (33%)
LOOSE: 67 (67%)
Player changes mind after one Joker is revealed
| Trial | Robert | Karen | Karen | Robert |
| 1 | zonk | win | win | zonk |
| 2 | win | win | zonk | zonk |
| 3 | win | win | win | zonk |
| 4 | win | win | win | win |
| 5 | zonk | win | zonk | zonk |
| 6 | zonk | win | win | win |
| 7 | win | zonk | win | win |
| 8 | win | zonk | zonk | win |
| 9 | zonk | zonk | win | win |
| 10 | win | zonk | win | win |
| 11 | win | win | win | win |
| 12 | win | win | zonk | win |
| 13 | win | zonk | win | zonk |
| 14 | win | win | win | win |
| 15 | zonk | win | zonk | zonk |
| 16 | win | zonk | win | win |
| 17 | win | win | zonk | win |
| 18 | win | zonk | win | win |
| 19 | win | win | zonk | win |
| 20 | zonk | zonk | win | win |
| 21 | zonk | win | win | zonk |
| 22 | win | zonk | win | win |
| 23 | win | win | win | win |
| 24 | win | zonk | win | win |
| 25 | win | zonk | zonk | win |
| TOTALS | ||||
| win | 18 | 14 | 17 | 19 |
| loose | 7 | 11 | 8 | 6 |
WIN: 68 (68%)
LOOSE: 32 (32%)
THE SURPRISE ENDING!
Changing your choice of doors after a "zonk" is revealed will double your chance of winning:
Or, you will win 2 out of 3 times if you choose to change your mind after the dealer reveals where the one of the goats is, because there is a 2 in 3 chance that you picked one of the goats to begin with.
What Monty Hall said...
May 12, 1975
Mr. Steve Selvin Asst. Professor of Biostatistics University of California, Berkeley
Dear Steve:
Thank you for sending me the problem from "The American Statistician."
Although I am not a student of a statistics problems, I do know that these figures can always be used to one's advantage, if I wished to manipulate same. The big hole in your argument of problems is that once the first box is seen to be empty, the contestant cannot exchange his box. So the problems still remain the same, don't they. . . one out of three. Oh, and incidentally, after one is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so. It was always two to one against him. And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.
Next time let's play on my home grounds. I graduated in chemistry and zoology. You want to know your chances of surviving with our polluted air and water?
Sincerely, Monty