Our last video explained the famous
Monty Hall problem. And from what I can
tell, reading your comments, there are
three types of people here: those who
understand the problem or even better
were helped by our video, those who still
just don't get it, and those who are
absolutely adamant that we got it wrong.
But one thing everyone has in common it
seems is a desire to discuss and explain
how they see the problem. I'm no
different, so what I'd like to do now is
quickly recap what the Monty Hall
problem is and then explain to you how I see it.
So we have these three doors: one hides
this shiny red sports car, the other two
have these worthless goats. You the
contestant choose a door, let's choose
door 1 in this example. The gameshow host,
Monty Hall himself, who knows what's
behind each door is then forced to
reveal a goat from behind one of the two
remaining doors. You're then given a
choice to stick or switch. With two doors
remaining it seems some of you think
this has become a 50/50 choice, stick or
switch, it doesn't really matter there
are two doors. One's got the car and one
doesn't; but that's not true and your
best chance of winning is to switch. Now
the reason for this is an important fact
that I cannot emphasise enough; Monty
Hall, the host, knows what's behind all
the doors and he's forced to show you a
goat from behind one of those two
remaining doors after your first choice.
This is very important. There is
information here that's about to be put
into the system, let me explain what
happens. When you made that first choice
there was a 1 in 3 chance that the car
was behind your door, surely we agree
here. Now in that situation Monty had two
goats he could have revealed to you from
behind these other two doors. Now that's
not very helpful, and of course if you
then switched like I tell you you should
you'd have moved away from the car and
lost. I'm sorry, I just lost you a sports
car, I apologise. But let's be honest,
there was only a 1 in 3 chance of
that happening. Let's wipe the slate
clean, or the brown paper clean, and now
this time let's imagine your first
choice was that dreaded goat. Now this is
twice as likely, there's a 2 in 3
chance that this is what you've done. We
agree on that right? 2 in 3 chance
you've chosen a goat. So this scenario of
course should be of more interest to us,
it's the more likely scenario. And this
time you have forced Monty's hand. He
knows where the one remaining goat is - he
knows it! And he has to show it to you. In
this case in two thirds of the games you
ever play he is indirectly telling you
exactly where the car is - it's here!
Behind the door he didn't open! Let me
say that again, in two thirds of the
games Monty can only open one door
because he knows, and therefore you
should know, the car is behind the other
one. And because you know this you should switch!
Now you don't know for sure this
has happened, there's a 1 in 3
chance this hasn't happened; but there's
a 2 in 3 chance it has happened,
it's more likely to have happened, it's
more probable. So there you go: switch!
Switch! Probability tells you to switch.
And this is because of Monty's knowledge;
if Monty didn't know what was behind the
doors this would be completely
different. Then after his big reveal
you'd be none the wiser and the odds
would indeed be 50-50, you may as
well stick. But it's also worth noting in
that case a third of the time Monty
would accidentally open a door that
reveals a car! That's not the game we're playing.
Monty knows where everything is and
won't reveal the car, that's the rules.
Monty has the knowledge and therefore
you have the knowledge; and knowledge, as
they say, is power. Leave a comment.