Welcome back to this module on applications of probability. The next problem is another playful application of probability the famous, some people may say infamous Monty Hall Problem. So what is this problem about? Congratulations! You're the winner of the games show, you've beat all the other candidates on this game show, it's a final moment for you to win a big prize a brand new car or go home with a goat. Here's how this Monty Hall problem works. You stand in front of three doors. Behind one of those doors is a brand new car that you can win. Behind other two doors is a goat behind each door. Now, you get to pick one door. Let's say you pick door one. Now the game show host gets to open one of the other doors. He is not allowed to open the door that you picked and he's not allowed to open the door with the big prize. He must open the door with the gold or with a goat. And now the game is back to you. So, he opens a door, let's say he opens door three and behind door three, is a goat. You pick door one. Now, you get to decide, you can keep door one or you can switch to door two. And then the prize is revealed. Either you're the lucky winner of a brand new car or the not so lucky winner of a goat. And before I tell you what to do here's the in-class quiz question for you: What should you do? Should you keep door one, door you already chose? Should you switch to the other remaining closed door? Or, it doesn't matter. So, what do you think? Did you think about this problem? Did you switch? Did you stay the same? Did you say it's fifty-fifty, it doesn't matter? Now, you should switch. The answer is that the probability of staying with the same door you have the probability of 1 in 3 to win. 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. There are 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 or that the objective definition that we learned about at the very beginning of this course. One in three, so the probability that you picked the right door, no matter which door it is. is one in three. There's a probability two in three that you're picking the wrong door. I think we can all agree on this. One in three, one third, 33.3% you are right two thirds you have the wrong door. Let's think about this now. The two thirds probability, that means that there are two doors where there's a prize but the prize can only be behind one door so either there's a prize behind door two and golden 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 car not two cars, there must be a goat among those doors that you did not choose. Now the game host, show master shows you that door and that goat. You just didn't learn anything. Nothing changed. The original probability was one third. The original probability was two thirds as it's behind one of the other doors and that included one goat. Nothing has changed. The game show host just made your life easy now because the two thirds is essentially on that one remaining door. And therefore you should switch. Many people find this very counter intuitive. So let me try to give you another sort of explanation. Let's think about, there isn't one door or three doors, but in fact there are a hundred doors Ninety nine goats, one car. You pick again, any door, let's say again door one. What's the probability that the car is behind that door? One in a hundred, very unlikely that you picked the right door. Now the game show host opens ninety eight doors. Now think about it. Ninety eight goats. Yeah, you knew there had to be 98 goats. Maybe 99 in the unlikely case you already have the car. Otherwise he can choose the 98 doors that he has to open with those dumb goats. So, would you switch now? Yeah, of course you want to switch, because there was one in a hundred before, it's going to be behind that other door most, most, most likely. So, I think that some of the confusion stems from that it is only three doors. If you think about more doors you will think: "Yeah, 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. And I'm now going to simulate this game a thousand 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 the probability definition number two so let's go to the spreadsheet. Here's now a spreadsheet where I simulate this Monty Hall game show problem for you. Let's have a look at of what I have done here. First, under price there's a ramp between one, two, three so this is a random number between one and three and that's a number of a door with the prize. So here in this example right now it's a two. The candidate chooses a two, again a random number. I assume that the candidate is completely clueless and just randomly picks a number. Then the host hast to open the 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 this here so this will hopefully run on anyone's spreadsheet on any type of computer, even older 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 looses. Now if I recalculate my sheet every time I do this a thousand times a thousand times I say: "Here's the prize, here's the candidate chooses this side, the host opens something and then we see what happens when 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 switch you win 66.1% of the time in these one thousand. Now let me click "recalculate sheet" notice the number changed. 34.6% not changing, 65% probability of winning yes, when you change. Here it's not 33.3% and 69.7%. So you see, our relative frequencies are indeed close to the one third, two third cut-off 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're ever in that situation please do me a favor switch that door. So, let's go back to the slides and I will tell you about some pop culture appearances of this problem. This concludes our look at the probabilities and the Monty Hall problem. As you can imagine, solving a cool problem like this that comes out of a game show makes it's 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 look at the clip, it's just two, 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" had 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, the every daily life or IQ related questions. Now when she explained this solution that you should switch the doors and then the probability is two thirds of winning she received a lot of hate mail and in particularly she received more than a thousand letters from PhDs in Mathematics who were ridiculing her and science saying that this is complete nonsense. It's a flip of a coin, fifty-fifty. 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 the true number is not fifty-fifty it's two-thirds-one-third and you should change. So, here I give you the link of Marilyn's website and there's a lot of entertaining discussion and a lot of these, sometimes hurtful and ridiculously offensive quotes from these Math PhDs. That brings us to the end of this second application after the birthday problem in the first lecture now we had the Monty Hall problem. This finishes our playful QT applications now we move on to applications from the business world. So please come back for more applications of probability. Thank you.