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.
Interesting! Thanks for the video! 99!!