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