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Studying for a test? Prepare with these 8 lessons on Probability and combinatorics.

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# The Monty Hall problem

Video transcript

Let's now tackle a
classic thought experiment in probability, called
the Monte Hall problem. And it's called the
Monty Hall problem because Monty Hall was the game
show host in Let's Make a Deal, where they would set up
a situation very similar to the Monte Hall problem
that we're about to say. So let's say that
on the show, you're presented with three curtains. So you're the contestant, this
little chef-looking character right over there. You're presented with
three curtains-- curtain number one, curtain number
two, and curtain number three. And you're told that behind
one of these three curtains, there's a fabulous prize,
something that you really want, a car, or a vacation, or some
type of large amount of cash. And then behind the
other two, and we don't know which they
are, there is something that you do not want. A new pet goat or an
ostrich or something like that, or a
beach ball, something that is not as good
as the cash prize. And so your goal is to try
to find the cash prize. And they say guess
which one, or which one would you like to select? And so let's say that you
select door number one, or curtain number one. Then the Monte
Hall and Let's Make a Deal crew will make it a
little bit more interesting. They want to show you
whether or not you won. They'll then show you one
of the other two doors, or one of the
other two curtains. And they'll show you one
of the other two curtains that does not have the prize. And no matter
which one you pick, there'll always be at
least one other curtain that does not have the prize. There might be two,
if you picked right. But there will always be
at least one other curtain that does not have the prize. And then they will
show it to you. And so let's say that they
show you curtain number three. And so curtain number
three has the goat. And then they'll
ask you, do you want to switch to curtain number two? And the question here is,
does it make a difference? Are you better off holding
fast, sticking to your guns, staying with the original guess? Are you better off switching
to whatever curtain is left? Or does it not matter? It's just random
probability, and it's not going to make a difference
whether you switch or not. So that is the brain teaser. Pause the video now. I encourage you
to think about it. In the next video, we will
start to analyze the solution a little bit deeper, whether
it makes any difference at all. So now I've assumed
that you've unpaused it. You've thought deeply about it. Perhaps you have
an opinion on it. Now let's work it
through a little bit. And at any point,
I encourage you to pause it and kind of
extrapolate beyond what I've already talked about. So let's think
about the game show from the show's point of view. So the show knows
where there's the goat and where there isn't the goat. So let's door number
one, door number two, and door number three. So let's say that our
prize is right over here. So our prize is the car. Our prize is the car, and
that we have a goat over here, and over here we
also have-- maybe we have two goats in
this situation. So what are we going
to do as the game show? Remember, the contestants
don't know this. We know this. So the contestant picks door
number one right over here. Then we can't lift
door number two because there's
a car back there. We're going to lift
door number three, and we're going to
expose this goat. In which case, it
probably would be good for the person to switch. If the person picks door number
two, then we as the game show can show either door number
one or door number three, and then it actually
does not make sense for the person to switch. If they picked
door number three, then we have to
show door number one because we can't
pick door number two. And in that case, it
actually makes a lot of sense for the
person to switch. Now, with that out of
the way, let's think about the probabilities
given the two strategies. So if you don't
switch, or another way to think about this strategy is
you always stick to your guns. You always stick to
your first guess. Well, in that situation, what
is your probability of winning? Well, there's three doors. The prize is equally likely
to be behind any one of them. So there's three possibilities. One has the outcome
that you desire. The probability of winning will
be 1/3 if you don't switch. Likewise, your
probability of losing, well, there's two ways
that you can lose out of three possibilities. It's going to be 2/3. And these are the
only possibilities, and these add up to
one right over here. So don't switch, 1/3
chance of winning. Now let's think about
the switching situation. So let's say always--
when you always switch. Let's think about how
this might play out. What is your
probability of winning? And before we even
think about that, think about how you would
win if you always switch. So if you pick wrong
the first time, they're going to show you this. And so you should always switch. So if you pick door
number one, they're going to show you
door number three. You should switch. If you picked wrong
door number three, they're going to show
you door number one. You should switch. So if you picked wrong and
switch, you will always win. Let me write this down. And this insight actually came
from one of the middle school students in the summer camp
that Khan Academy was running. It's actually a fabulous
way to think about this. So if you pick
wrong, so step one, so initial pick is wrong,
so you pick one of the two wrong doors, and then step
two, you always switch, you will land on the car. Because if you picked
one of the wrong doors, they're going to have to
show the other wrong door. And so if you
switch, you're going to end up on the right answer. So what is the probability of
winning if you always switch? Well, it's going to be the
probability that you initially picked wrong. Well, what's the probability
that you initially picked wrong? Well, there's two
out of the three ways to initially pick wrong. So you actually have a
2/3 chance of winning. There's a 2/3 chance you're
going to pick wrong and then switch into the right one. Likewise, what's
your probability of losing, given that you're
always going to switch? Well, the way that you would
lose is you pick right, you picked correctly. In step two, they're
going to show one of the two empty
or non-prize doors. And then step
three, you're going to switch into the
other empty door. But either way, you're
definitely going to switch. So the only way to lose, if
you're always going to switch, is to pick the right
the first time. Well, what's the
probability of you picking right the first time? Well, that is only 1/3. So you see here, it's
sometimes counter-intuitive, but hopefully this makes
sense as to why it isn't. You have a 1/3 chance of winning
if you stick to your guns, and a 2/3 chance of winning
if you always switch. Another way to
think about it is, when you first make
your initial pick, there's a 1/3 chance
that it's there, and there's a 2/3
chance that it's in one of the other two doors. And they're going to
empty out one of them. So when you switch,
you essentially are capturing that
2/3 probability. And we see that right there. So hopefully you enjoyed that.