The Monty Hall Problem
Let's take a look at one of the most counter-intuitive problems in the world and see what we can learn from it:
Imagine you are on one of those TV game shows where you have to choose one of three curtains. Behind one curtain is a wonderful prize, behind the other two curtains – nothing. You choose a curtain and then the host opens one of the curtains you did not choose to show you there is no prize behind it. Then he says you can keep the curtain you originally chose or switch to the other curtain that is still closed.
What is your best strategy for winning the prize?:
- Keep your first choice?
- Switch to the other closed curtain?
- Or does it make no difference either way?
This is a famous problem that Marilyn Vos Savant wrote about in her column in Parade Magazine back in 1990. That's when I first read about it.
Marilyn was known for years as the person with the highest IQ in the world but when she gave her answer to this problem thousands of people wrote to her to say she was wrong. Mathematicians with Ph.D's wrote to her to say she was wrong. Even one of the most famous mathematicians in the world thought she was wrong.
But she wasn't wrong....
She was right....
She said the answer was B and that is demonstrably true. What makes this problem so interesting is that so many smart people think the answer is C and then can't even understand WHY the right answer is B even after B is proven to BE the right answer.
This is a wonderful problem that can teach us something about how we think, how our thinking sometimes goes wrong, and maybe even something about the deep nature of reality itself.
But first, how can we show that the answer really is B?
First Explanation
Think about it like this:
When you first chose a curtain you had a 1/3 chance of winning and a 2/3 chance of losing.
If the host then opens one of the losing curtains and asks you if you want to switch your choice we know this:
- if you originally chose the winning curtain then switching will make you lose,
- but if you originally chose a losing curtain then switching will make you win.
Since you originally had a 1/3 chance of picking the winning curtain then switching gives you a 1/3 chance of losing.
Since you originally had a 2/3 chance of picking a losing curtain then switching gives you a 2/3 chance of winning.
You are twice as likely to win this game if you switch curtains!
Second Explanation
Let's try analyzing the Monty Hall problem another way: by simply considering every option and counting up how many ways there are to win and how many ways there are to lose.
We'll look at every option when the prize is behind curtain #1. (If you were to repeat this analysis assuming the prize is behind curtain #2 or curtain #3 the results would be the same.)
OK... the prize is behind curtain #1 but, of course, you don't know that when you are playing the game.
- If you pick curtain #1 and do not switch then you win.
- If you pick curtain #2 and do not switch then you lose.
- If you pick curtain #3 and do not switch then you lose.
By following the "do not switch" strategy there is ONE way to win and TWO ways to lose. Therefore, if you make your original choice of curtain at random and then refuse to switch you have a 1/3 chance of getting the prize and a 2/3 chance of not getting the prize.
Now let's see what happens if we follow the switching strategy....
- If you pick curtain #1 the host will show you what's behind either curtain #2 or curtain #3, then you will switch to the remaining closed curtain and lose.
- If you pick curtain #2 the host will show you what's behind curtain #3, then you will switch to curtain #1 and win.
- If you pick curtain #3 the host will show you what's behind curtain #2, then you will switch to curtain #1 and win.
By following the switching strategy there are TWO ways to win and ONE way to lose. If you make your original choice of curtain at random, and then switch, you have a 2/3 chance of winning and only a 1/3 chance of losing.
This is one of the most counter-intuitive problems EVER CONCEIVED but I hope this little thought experiment will persuade you that the counter-intuitive result really is true!
Third Explanation
The amazing thing about this seeming paradox is that the CHOICE you make at the beginning affects what happens to the probabilities later. Let's consider an alternative form of the problem with 100 curtains and see if that makes it more clear.
In this version of the game there are 100 curtains. Behind one of those curtains is a prize and behind the other 99 curtains there is nothing.
You choose one curtain at random. I think everyone will agree that your chance of winning right now is 1%.
By making a choice you are dividing the curtains into two sets: the set you CHOSE, which has one curtain in it, has a 1% of being the winner and the set you did NOT choose, which has 99 curtains in it, has a 99% chance of having the winner. In the set you did not choose, the 99% chance of having the winner is divided between 99 curtains, so each of those curtain has a 1% chance of being the winner.
Now suppose the game show host opens 49 of the curtains you did not choose, to show you those 49 curtains have no prize behind them. That leaves 50 closed curtains in the set you did not choose but that set must still have a 99% chance of having the winner, so each of those curtains now has a 1.98% chance of winning. (99% chance of winning divided by 50 curtains.) The set you chose still has one curtain with a 1% chance of winning.
Next, imagine the game show host opens 40 more curtains in the set you did not choose and shows you that the prize is not behind any of those. Now there are just 10 curtains left in the set you did NOT choose but those 10 curtains must STILL share a 99% chance of having the winner so each of them has a 9.9% chance of being the winner.
Finally, imagine the game show host continuing to open curtains in the set you did not choose, and showing you that:
there is no prize behind this one,
or this one,
or this one...,
until there is only one curtain left. Since there was a 99% chance the prize was in the set of curtains you did not choose, and since there is now only one curtain left in that set, then the chance of that curtain having the prize behind it is 99%. The chance of the prize being behind the curtain you originally chose is still just 1%. Obviously, if you have the chance, and if you want to win, you should switch curtains.
Mystical Math and Lessons Learned
As we have said, this problem or paradox we have been discussing is so counter-intutive that even professional mathematicians have trouble accepting it as true. I hope that one or the other of the three explanations given above will help you to see that the strange result is true even if it remains strange!
Just to stress the strangeness let's think back to the third explanation given above and imagine a second contestant coming in right at the end when 98 curtains have been opened and you, contestant 1, are trying to decide whether to switch curtains or not.
This contestant 2 is told that we started with 100 curtains, one with a prize and 99 without a prize. He or she is told that the host opened 98 of the curtains with no prize so that now there are just two curtains left unopened. Contestant 2 is now asked to calculate the probabilities of the prize being behind one curtain or the other. Based on everything contestant 2 knows he says the probability is 50% for each curtain. That is is absolutely correct!
But then we ask you, contestant 1, to calculate the probabilities and you say that the probability for the curtain you originally chose is 1% and the probability for the one remaining curtain that you did not choose is 99%. That is also absolutely correct!
How can two people looking at the same two curtains come up with such radically different probabilities? What is different about contestant 1 and contestant 2 that leads to such a radical difference in the probabilities they CORRECTLY calculate?
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[Check out this book about mathematical paradoxes. I haven't read it yet so if you get to it first please write a little review I can publish here.]
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The difference is that contestant 1 started the game by CHOOSING one option and contestant 2 did not. It was a human choice that resulted in such a divergent calculation of probabilities later on. In a way, it was a human choice that changed reality in this problem. Human choices changing reality... that seems kind of MYSTICAL to me.
Finally, what should we learn from this seeming paradox? Most people who hear this problem for the first time, even experts, assume that it makes no difference whether you switch curtains or not. They are confident of that answer; they are sure they are right - but they are wrong!
If people can be wrong about an easy-to-describe problem like this how often do you think people might be wrong about complex political issues, and economic problems, and ethical disputes, and court cases on constitutional issues? How often must we be wrong about questions of social injustice, or war and peace, or how to grow the economy, or how to save the environment?
I'm not trying to promote skepticism here! Even though people are often SURE and just as often WRONG we can NEVER give up trying to improve our thinking and our understanding of truth. We need to think hard and make the best choices we can at each moment of our lives but THEN we have to keep going back to look at that issue again and again to see if we notice something we didn't notice before.
Don't make up your mind once and for all!
Especially when someone says we are wrong and they tell us they can explain why, we should stop and listen, and see if they have some new knowledge to impart to us.
Keep thinking! Keep learning! Never stop!
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Copyright © 2017 by Joseph Wayne Gadway
