Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors Monty Hall opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door i. The Monty Hall problem is deciding whether you do.
The correct answer is that you do want to switch. The problem can be generalized to four doors as follows. Let one door conceal the car, with goats behind the other three.
Pick a door. Then the host will open one of the nonwinners and give you the option of switching. Call your new choice which could be the same as if you don't switch. The host will then open a second nonwinner, and you must decide for choice if you want to stick to or switch to the remaining door. The probabilities of winning are shown below for the four possible strategies. The above results are characteristic of the best strategy for the -stage Monty Hall problem: stick until the last choice, then switch.
Barbeau, E. Bogomolny, A. Dewdney, A. New York: Wiley, Donovan, D. Ellis, K. Flannery, S. Here are some examples:. You blew it, and you blew it big! Whether you change your selection or not, the odds are the same. I am sure you will receive many letters on this topic from high school and college students.
Perhaps you should keep a few addresses for help with future columns. Maybe women look at math problems differently than men. You should switch. There are three possibilities for where the car could have been placed. If the car is behind Door 1 top left , you win. If the car is behind Door 2 middle left , you lose. If the car is behind Door 3 bottom left , you lose. If the car is behind Door 1, you lose. If the car is behind Door 2, Monty would have opened Door 3, so you would switch to Door 2 and win.
If the car is behind Door 3, he would have opened Door 2, so you would switch to Door 3 and win. Nowadays, you can play it online. Many people insist that each of the unknown alternatives in this case, the unopened doors must have an equal probability. That is true of symmetrical gambling toys like the faces of a coin or sides of a die, and it is a reasonable starting point when you know absolutely nothing about the alternatives.
But it is not a law of nature. Certainly there were failures of critical thinking coming from sexism, ad hominem biases, and professional jealousy. She defied the stereotype of a mathematician, and her celebrity and bragging rights from Guinness made her a big fat target for a takedown.
But part of the problem is the problem itself. Many persist even when they see it simulated and even when they repeatedly play for money. Many visualize the causal chain. You might get 2 heads in a row and think it was rigged.
Just play the game a few dozen times to even it out and reduce the noise. The best I can do with my original choice is 1 in 3. Instead of the regular game, imagine this variant:. Try this in the simulator game; use 10 doors instead of Your decision: Do you want a random door out of initial guess or the best door out of 99?
Said another way, do you want 1 random chance or the best of 99 random chances? Filtered is better. Yes, two choices are equally likely when you know nothing about either choice.
You pick the name that sounds cooler, and is the best you can do. You know nothing about the situation. Would this change your guess? Your uninformed friend would still call it a situation. With the Japanese baseball players, you know more than your friend and have better chances. The more you test the old standard, the less likely the new choice beats it. This is what happens with the door game. The odds are the champ is better than the new door, too. At the start, every door has an equal chance — I imagine a pale green cloud, evenly distributed among all the doors.
On and on it goes — and the remaining doors get a brighter green cloud. He is purposefully not examining your door and trying to get rid of the goats there.
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