2:33

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

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

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

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

Â They have three doors, you have no information, so

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

Â That's wonderful.

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

Â 3:32

Let's think about this now.

Â So two-thirds probability,

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

Â But the price can only be behind one door.

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

Â gold in door three, or the other way around.

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

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

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

Â 5:06

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

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

Â You knew there had to be 98 goats,

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

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

Â So would you switch now?

Â Yeah, of course, you want to switch.

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

Â other door most, most, most likely.

Â 5:42

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

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

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

Â simulation of this problem in a spreadsheet.

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

Â And from the simulation create data,

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

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

Â So now, because we're looking at data,

Â we're back to probability definition number two.

Â So let's go to the spreadsheet.

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

Â show problem for you.

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

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

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

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

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

Â The candidate chooses a 2.

Â Again, a random number.

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

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

Â Now here, this is quite a complicated Excel formula.

Â You can ignore this, I know.

Â Many of you can code this very elegantly and

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

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

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

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

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

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

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

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

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

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

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

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

Â Now, let me click recalculate sheet.

Â Notice the numbers change.

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

Â Here's now 30.3%, 69.7.

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

Â two-third cutoff that I explained to you earlier.

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

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

Â the probability of winning after switching is twice as high

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

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

Â 9:22

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

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

Â a game show makes its way into popular culture.

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

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

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

Â It's just two and a half minutes.

Â I think you will enjoy it.

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

Â Namely a reader of a weekend magazine called Parade,

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

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

Â And she answers all kinds of questions that people have,

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

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

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

Â And in particular, she received more than 1,000

Â letters from PHDs in mathematics who were ridiculing her answer.

Â And said this is complete nonsense.

Â It's a flip of a coin, 50/50.

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

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

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

Â discussion where true number is not 50/50.

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

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

Â And there's a lot of entertaining discussion.

Â And a lot of these, sometimes hurtful and

Â ridiculously offensive quotes from these math pages.

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

Â the first lecture.

Â Now, we had the module problem.

Â This finishes our playful cutie applications,

Â now we move on to applications from the business world.

Â So please come back for more applications of probability.

Â Thank you.

Â