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The Monty Hall Problem: A really short explanation
Today I was trying to explain the Monty Hall Problem to some friends at the gaming store, and I came up with a simpler explanation than I've used in the past, so I figured I'd share.
For those not familiar with it, here's the problem: there are three doors, behind one is a prize and behind the other two are nothing (well, gag prizes in the actual Let's Make A Deal show, but essentially nothing). You pick one of the doors. Monty opens one of the doors you didn't pick, and gives you the chance to switch to the unopened door you didn't pick. Should you? (Note: in the real show, he doesn't always offer a switch, and may be more likely to offer if your pick was correct in the first place, but we're ignoring that complication.)
The answer, which can be explained in probability terms, quantum states, information theory, etc, is that you should. This is, of course, counterintuitive and messes with people's brains, as it has ever since Marilyn vos Savant first claimed it. After all, once a door is opened, each remaining door has a 50% chance of being right, yes? So why change?
Here's a simple explanation. Let's change the situation slightly. What if, after picking one door, Monty doesn't open any doors, but just offers you the contents of BOTH of the other doors? Obviously, you would take it, right? Two doors is twice as good as one door, even if you know for sure that one of those two doors has nothing behind it.
Well, that's basically what he does anyway. He offers you the "nothing" behind the door he does open, plus whatever is behind the unopened door. So, you pick #2, he opens #1 to reveal a goat or something, and offers you #3. But he's really offering you both #1 and #3, which is two out of the three possibilities, not one out of two.
Of course, as mentioned earlier, he doesn't always open a door. And it's possible he's more likely to open a door if you were right in the first place, balancing out the overall odds. But in a case where he doesn't know which door is correct before making the offer (presumably someone who does know which is the prize picks which to open once he makes the offer), it's best to switch.
Now, Deal or No Deal works on a similar principle, but is MUCH more mathematically complicated, and includes a multiplier to the expectation value of the remaining prize amounts based on how many deals have been offered already....
For those not familiar with it, here's the problem: there are three doors, behind one is a prize and behind the other two are nothing (well, gag prizes in the actual Let's Make A Deal show, but essentially nothing). You pick one of the doors. Monty opens one of the doors you didn't pick, and gives you the chance to switch to the unopened door you didn't pick. Should you? (Note: in the real show, he doesn't always offer a switch, and may be more likely to offer if your pick was correct in the first place, but we're ignoring that complication.)
The answer, which can be explained in probability terms, quantum states, information theory, etc, is that you should. This is, of course, counterintuitive and messes with people's brains, as it has ever since Marilyn vos Savant first claimed it. After all, once a door is opened, each remaining door has a 50% chance of being right, yes? So why change?
Here's a simple explanation. Let's change the situation slightly. What if, after picking one door, Monty doesn't open any doors, but just offers you the contents of BOTH of the other doors? Obviously, you would take it, right? Two doors is twice as good as one door, even if you know for sure that one of those two doors has nothing behind it.
Well, that's basically what he does anyway. He offers you the "nothing" behind the door he does open, plus whatever is behind the unopened door. So, you pick #2, he opens #1 to reveal a goat or something, and offers you #3. But he's really offering you both #1 and #3, which is two out of the three possibilities, not one out of two.
Of course, as mentioned earlier, he doesn't always open a door. And it's possible he's more likely to open a door if you were right in the first place, balancing out the overall odds. But in a case where he doesn't know which door is correct before making the offer (presumably someone who does know which is the prize picks which to open once he makes the offer), it's best to switch.
Now, Deal or No Deal works on a similar principle, but is MUCH more mathematically complicated, and includes a multiplier to the expectation value of the remaining prize amounts based on how many deals have been offered already....