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Monty Hall Problem - Numberphile - YouTube
Channel: Numberphile
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Well 'Let's Make a Deal' was a popular
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show back in the day; contestants could
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go on this game show and maybe go home
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with the car of their dreams. First of
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all the game show host was a very famous
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guy named Monty Hall. And Monty would
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come on to the show and he would have
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three doors. And a contestant, that's you,
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would come on to the show, Monty would
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give you the chance to choose door
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number 1, door number 2 or door
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number 3. Now behind one of those
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doors, only one, was your dream flash car.
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Bright red, sports car, very fast; it was
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awesome. And behind the other two doors
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were 'zonks', that was the
'Let's Make a Deal'
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word for something that you don't really
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want. So you had- looks like a 1 in
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3 chance of getting your dream car.
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So suppose you pick door number 1; and
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Monty would then do the same thing every
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week. He would go over to the two doors
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you didn't pick, 2 and 3, and he
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would open one of them. Let's suppose he
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opened door number 2. And the door that
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Monty opened would always have behind it
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a zonk.
- (So- so he knew?)
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Well he knows everything
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right? He's the game show host. And then
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Monty looked at you in the face and say,
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'do you like door number 1 or do you
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want to switch?' There's only one thing to
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switch to, in this case it's door number
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3. Are you gonna stay with your
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choice, or are you gonna make that leap
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to something different? And very often
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contestants would stand there agonised,
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right? So they've got some new
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information-
- (So stick or switch?)
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Stick or switch. Well, a
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remarkable thing about this problem,
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simple as it is, is that it has sparked
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just endless debate. In the time of the
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show I don't recall anybody ever saying
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there was a dedicated strategy that you
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should always follow, but in fact there
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is a dedicated strategy should follow:
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you should pick door number 3. That's
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the answer, you should switch. You should
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switch every time and that will do the
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best for you over the long run. So there
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is a 1/3 chance that the car is
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behind the door you picked initially,
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that means there must be a 2/3 chance,
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much greater twice as big, that the car is
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somewhere else. And since we know that
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somewhere else cannot be door number 2,
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because Monty showed us that, it's got to
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be over here. So this is what you should
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choose, you should switch. Twice as likely
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to have the car behind the door that you
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didn't pick as the door that you did. 2/3
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probability to a 1/3 probability. Now if
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you switch are you guaranteed to win?
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Absolutely not. But if you play this game
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over and over again, on average, you will
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win 2/3 of the time so switch is your
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strategy and you can't beat it. One thing
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that you might say is that the initial
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2/3 chance that the car was behind door
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number 2 and door number 3 got
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concentrated behind the door that Monty
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did not open. That's effectively what's
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happening, that's intuitively what's
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happening, and that in fact is what the
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mathematics shows is happening. Now there
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is a way to see that in kind of a more
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grand way; if we imagine not having three
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doors but we imagine having a hundred
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doors. And let's imagine that we're
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playing the same kind of game, we have
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Monty over here, he is going to give you
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an opportunity to pick a door and your
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dream flash car is behind one of these
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hundred doors but there are 99 zonks
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all behind the other doors. Now what is
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it that you're going to do? Well you're
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going to pick a door, say again you pick
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door number 1. Now you're feeling a
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little bit different than you might have
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felt in the case where there were three
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doors; because there you thought, wow I've
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got a 1 in 3 chance. Here you're
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thinking I've got a 1 in 100 chance,
I'm not
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gonna get that car. It's not behind door
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number 1, it's probably behind one of
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the other 99 doors. What Monty does is he
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opens 98 of those 99 doors, he shows you
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98 zonks and he asked you now, do you
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want to switch? Well maybe just because
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of the sheer numbers this thing is a
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little clearer. You know that there was
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99 out of a 100 chance that
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the car was over here and now the only
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door you're left with after Monty
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shows you all those zonks
is door, say, number
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37. And you're thinking, this is no good,
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this is no good, all those doors are no
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good; do I want to stay with 1 or do I
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want to stay with 37? You can sort of
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almost feel the concentration of that 99
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percent going behind door number 37 and
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so you switch over here and very likely
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you're gonna get that car and drive away.
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(I like your car drawing.)
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Well thank you!
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(That makes more sense doesn't it?
Suddenly it)
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(seems like a no-brainer.)
- It does seem
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like a no-brainer but in fact when it's
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on a smaller scale, maybe just because
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1/3 and 2/3 are a whole lot closer to
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each other than 1 in 100 and
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99 in a 100, that the the
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point is is obscured.
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[Preview] X and Y is the probability
of X given
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that Y happens, given that you know you
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open door number 2, what's the
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probability of the cars there, times the
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probability of Y happening by itself.
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