2:33

The probability of switching to the other door, is two in three.

So you can double your chances of winning by switching to the other door.

Let's think about why that's the case.

Let's go back to the beginning of the problem.

They have three doors, you have no information, so

all doors, I think we can agree, are equally likely.

That's wonderful.

We can use, the definition number one, the classical probability definition.

3:32

Let's think about this now.

So two-thirds probability,

that means there are two doors, where there's a price.

But the price can only be behind one door.

So there's either the price is behind door two and

gold in door three, or the other way around.

So the two-thirds probability includes a fact that behind one door

there must be a goat because there's only one cow not two cows.

There must be a goat among those doors that you did not choose.

5:06

1 in 100, very unlikely that you pick the right door.

Now, the game show host opens 98 doors, now think about it 98 goats.

You knew there had to be 98 goats,

maybe 99 in the unlikely case you already at the cat otherwise.

He can choose at 98 doors that he has to open with those damn goats.

So would you switch now?

Yeah, of course, you want to switch.

Because you have most one in a hundred before it's going to be behind that

other door most, most, most likely.

5:42

So I think some of the confusion stems from that is only three goats.

If you think about more goats, you will think I definitely want to switch.

I know this is very tricky, so now let's look at a Monte Carlo

simulation of this problem in a spreadsheet.

I'm now going to simulate this game 1,000 times.

And from the simulation create data,

create proportions of winning by switching or winning by not switching.

And then, let's have a look at those numbers.

So now, because we're looking at data,

we're back to probability definition number two.

So let's go to the spreadsheet.

Here's now a scratch sheet where I simulate this Monty Hall game

show problem for you.

Let's have a look at what I've done here.

First, prize this rand between 1, 2, 3.

So this is a random number between 1 and 3, and

that's the number of the door where the prize is.

So here, in this example right now, it's a 2.

The candidate chooses a 2.

Again, a random number.

I assume that the candidate is completely clueless and just randomly picks a number,

then the host has to open a door with a goat.

Now here, this is quite a complicated Excel formula.

You can ignore this, I know.

Many of you can code this very elegantly and

much faster than I did using Excel Macros or Visual Basic.

I deliberately did not do the CSO, this will hopefully, run on anyone's

spreadsheet on any type of computer, even all your computers in the world.

Then, we look at what happens if the candidate does not switch,

keeps the same door, or what happens if the candidate switches.

And then, here we see with a simple if, question whether he or she wins or loses.

Now, if I recalculate my sheet, every time I do this 1,000 times,

1,000 times I say, he is a prize, he is a candidate, he chooses this site, the host

open something and then we see what happens with switching or not switching.

Here, for example, right now on my sheet it says,

if you do not switch, you win 33.9% of the time.

If you yes, you switch, you win 66.1% of the time in these 1,000.

Now, let me click recalculate sheet.

Notice the numbers change.

34.6% not changing, 65% probability of winning, yes when you change.

Here's now 30.3%, 69.7.

So you see, our relative frequencies are indeed close to the one-third,

two-third cutoff that I explained to you earlier.

We can't expect that we hit it exactly, but we get very close.

And I hope this convinces the last doubters among you that indeed,

the probability of winning after switching is twice as high

as the probability of winning when you don't switch.

So if you ever in that situation, please do me a favor, switch that door.

9:22

This concludes our look at the probabilities in the Monty Hall problem.

As you can imagine, sort of a cute problem like this that comes out of

a game show makes its way into popular culture.

In the movie 21, there's a really cute scene where a student at MIT

has to explain this problem and the solution to his professor.

Please Google it, and have a look at the clip.

It's just two and a half minutes.

I think you will enjoy it.

In 1990, there was a lot of controversy about this problem.

Namely a reader of a weekend magazine called Parade,

has sent in this question to a column called Ask Marilyn.

This woman Marilyn is supposedly the highest IQ person on the globe.

And she answers all kinds of questions that people have,

whether it's in their love life, in their everyday life, or IQ-related questions.

Now, when she explained this solution, and that you should switch the doors and

that the probability then is two-thirds of winning, she received a lot of hate mail.

And in particular, she received more than 1,000

letters from PHDs in mathematics who were ridiculing her answer.

And said this is complete nonsense.

It's a flip of a coin, 50/50.

Now, all of them later on had to take their criticism and their laughter back

when she then, in great detail, convinced them of the right answer.

And as we just saw on the Monte Carlo simulation I think there can be no

discussion where true number is not 50/50.

It's two-thirds, one-third and you should change.

So here I give you the link of a Marylyn's website.

And there's a lot of entertaining discussion.

And a lot of these, sometimes hurtful and

ridiculously offensive quotes from these math pages.

That brings us to the end of this second application after the birthday problem in

the first lecture.

Now, we had the module problem.

This finishes our playful cutie applications,

now we move on to applications from the business world.

So please come back for more applications of probability.

Thank you.