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| The Monty Hall Paradox |
The Monty Hall Paradox
There was a popular television game show in the 1970’s called ‘Let’s Make a Deal’ with host Monty Hall. In a regular segment of the show, a contestant was shown three doors and was given the details of a really good prize that was behind one door, and poor consolation prizes behind the other two. With a lot of showmanship, the typical game then proceeded as follows:
1. The contestant selected Door #1, Door #2, or Door #3, hoping the prize was behind that door. The door was not opened at this stage of the game.
2. Monty then opened one of the other two doors to reveal a consolation prize.
3. The contestant was then given the opportunity to switch from the original selection to the third and last door.
The problem is to determine whether or not switching doors would have been a good strategy. This is a problem in deduction. The key to its solution is to find a way to express the components of the problem so that the implications are more obvious.
There was a popular television game show in the 1970’s called ‘Let’s Make a Deal’ with host Monty Hall. In a regular segment of the show, a contestant was shown three doors and was given the details of a really good prize that was behind one door, and poor consolation prizes behind the other two. With a lot of showmanship, the typical game then proceeded as follows:
1. The contestant selected Door #1, Door #2, or Door #3, hoping the prize was behind that door. The door was not opened at this stage of the game.
2. Monty then opened one of the other two doors to reveal a consolation prize.
3. The contestant was then given the opportunity to switch from the original selection to the third and last door.
The problem is to determine whether or not switching doors would have been a good strategy. This is a problem in deduction. The key to its solution is to find a way to express the components of the problem so that the implications are more obvious.
At a first glance, the following statements seem to be correct:
1. Since there are three doors and your selection is essentially random, the probability of winning the grand prize after your first selection should be one in three.
2. After Monty reveals a consolation prize, there are only two doors left, so your chances of winning should increase to one in two.
3. There appears to be no obvious advantage to switching your choice; there are still two doors with a big prize behind only one door.
In reality, participants who switched doors won about two times out of three. That is the paradox. How could switching doors be an advantage? If you search the Internet using the key words ‘Monty Hall paradox’ you will find hundreds of sites that explore this problem.
To simplify the discussion, let’s assume that the prize is behind Door #1, but as a contestant you do not know that. You still have a random choice of one of the three doors. As Monty takes his turn he will always open one of the two remaining doors to reveal a consolation prize, never the door to the grand prize.
Working through the possibilities:
1. If you picked Door #1 – then Monty will open either Door #2 or Door #3. In this case, if you switch from Door #1 you lose.
2. If you picked Door #2 – since Monty knows the prize is behind Door #1, he has to open Door #3 revealing a consolation prize. In this case if you switch to Door #1 you win.
3. If you picked Door #3 – then similarly Monty has to open Door #2 revealing another consolation prize. Again, in this case if you switch to Door #1 you win.
The net result is that if the prize is behind Door #1 and your first choice is random, then two times out of three, you will win the grand prize if you switch doors after Monty reveals a consolation prize. The same pattern applies if the prize is behind Door #2 or Door #3.
The key to the paradox is the phrase Monty ‘has to open’ in the second and third possibilities. For example, if the prize is behind Door #1 and you have selected Door #2, then Door #3 is the only choice available for Monty to open without spoiling the game. In two cases out of three, your initial choice plus Monty’s forced choice direct you to the winning door.
1. Since there are three doors and your selection is essentially random, the probability of winning the grand prize after your first selection should be one in three.
2. After Monty reveals a consolation prize, there are only two doors left, so your chances of winning should increase to one in two.
3. There appears to be no obvious advantage to switching your choice; there are still two doors with a big prize behind only one door.
In reality, participants who switched doors won about two times out of three. That is the paradox. How could switching doors be an advantage? If you search the Internet using the key words ‘Monty Hall paradox’ you will find hundreds of sites that explore this problem.
To simplify the discussion, let’s assume that the prize is behind Door #1, but as a contestant you do not know that. You still have a random choice of one of the three doors. As Monty takes his turn he will always open one of the two remaining doors to reveal a consolation prize, never the door to the grand prize.
Working through the possibilities:
1. If you picked Door #1 – then Monty will open either Door #2 or Door #3. In this case, if you switch from Door #1 you lose.
2. If you picked Door #2 – since Monty knows the prize is behind Door #1, he has to open Door #3 revealing a consolation prize. In this case if you switch to Door #1 you win.
3. If you picked Door #3 – then similarly Monty has to open Door #2 revealing another consolation prize. Again, in this case if you switch to Door #1 you win.
The net result is that if the prize is behind Door #1 and your first choice is random, then two times out of three, you will win the grand prize if you switch doors after Monty reveals a consolation prize. The same pattern applies if the prize is behind Door #2 or Door #3.
The key to the paradox is the phrase Monty ‘has to open’ in the second and third possibilities. For example, if the prize is behind Door #1 and you have selected Door #2, then Door #3 is the only choice available for Monty to open without spoiling the game. In two cases out of three, your initial choice plus Monty’s forced choice direct you to the winning door.
