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Mathematical Paradoxes: The Monty Hall problem
It has been some time since the first mathematical “paradox” I posted: the twin coin flipping which generated lots of discussion and controversy! So, with no further delay, I am moving on to another classic: the Monty Hall problem.
In a reality show you are presented with three doors Red, Green, and Blue. One of the doors contains one million dollars as a prize! Let us say you choose the Red door. The presenter will not open the door immediately. Instead the presenter (who knows where the prize is) must first open one of the other two doors, Green or Blue, but not the one with the prize (in case the one you picked doesn’t contain the prize). So, he opens the Green door and there is no prize behind it. He then asks if you would like to change your original selection (Red door) and choose the Blue door instead. What do you do? Do you stick to your initial selection (the Red door) or do you change for the Blue one?
In a reality show you are presented with three doors Red, Green, and Blue. One of the doors contains one million dollars as a prize! Let us say you choose the Red door. The presenter will not open the door immediately. Instead the presenter (who knows where the prize is) must first open one of the other two doors, Green or Blue, but not the one with the prize (in case the one you picked doesn’t contain the prize). So, he opens the Green door and there is no prize behind it. He then asks if you would like to change your original selection (Red door) and choose the Blue door instead. What do you do? Do you stick to your initial selection (the Red door) or do you change for the Blue one?
Not very good at maths but methinks it wouldnt make any difference
2. Jonathan
The probability of being behind my door is 33%. The same for the other doors. So in the end it will be the same probability whichever door I choose
3. Chris
You should now choose the blue door.
At the outset you had a 1/3 chance of getting it right (with the red door) and there was a 2/3 probability that either the green or blue door had the prize. Opening the green door doesn’t change those probabilities. So (after the green door has been opened) the odds of the prize being behind the blue door are now 2/3.
At the outset you had a 1/3 chance of getting it right (with the red door) and there was a 2/3 probability that either the green or blue door had the prize. Opening the green door doesn’t change those probabilities. So (after the green door has been opened) the odds of the prize being behind the blue door are now 2/3.
4. Chris
I don’t really like my last reply as it relies too much on probability and not enough on logic and possibly expects too much from the reader. So -
The prize is definitely inside one of the doors and it stays there. There are only three possibilities:
1: The prize is behind the red door. The host can open the green or blue door. It doesn’t matter which, the unopened door doesn’t contain the prize either.
2: The prize is behind the green door. The host must open the blue door.
3: The prize is behind the blue door. The host must open the green door.
Only in case 1 has the unopened door not got the prize behind it. In the other two cases, the unopened door has got the prize behind it.
All three cases are equally likely, with probability 1/3. Therefore, the probability of initially selecting the prize door is 1/3. The probability of the unopened door containing the prize is 2/3 (i.e. in 2 out of 3 cases the unopened box contains the prize).
So you should swap to the unopened door (the blue one in this particular case).
The prize is definitely inside one of the doors and it stays there. There are only three possibilities:
1: The prize is behind the red door. The host can open the green or blue door. It doesn’t matter which, the unopened door doesn’t contain the prize either.
2: The prize is behind the green door. The host must open the blue door.
3: The prize is behind the blue door. The host must open the green door.
Only in case 1 has the unopened door not got the prize behind it. In the other two cases, the unopened door has got the prize behind it.
All three cases are equally likely, with probability 1/3. Therefore, the probability of initially selecting the prize door is 1/3. The probability of the unopened door containing the prize is 2/3 (i.e. in 2 out of 3 cases the unopened box contains the prize).
So you should swap to the unopened door (the blue one in this particular case).
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