Basic theoretical probability
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The Monty Hall problem
Let's now tackle a classic thought experiment in probability, called the Monte Hall problem. And it's called the Monty Hall problem because Monty Hall was the game show host in Let's Make a Deal, where they would set up a situation very similar to the Monte Hall problem that we're about to say. So let's say that on the show, you're presented with three curtains. So you're the contestant, this little chef-looking character right over there. You're presented with three curtains-- curtain number one, curtain number two, and curtain number three. And you're told that behind one of these three curtains, there's a fabulous prize, something that you really want, a car, or a vacation, or some type of large amount of cash. And then behind the other two, and we don't know which they are, there is something that you do not want. A new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize. And so your goal is to try to find the cash prize. And they say guess which one, or which one would you like to select? And so let's say that you select door number one, or curtain number one. Then the Monte Hall and Let's Make a Deal crew will make it a little bit more interesting. They want to show you whether or not you won. They'll then show you one of the other two doors, or one of the other two curtains. And they'll show you one of the other two curtains that does not have the prize. And no matter which one you pick, there'll always be at least one other curtain that does not have the prize. There might be two, if you picked right. But there will always be at least one other curtain that does not have the prize. And then they will show it to you. And so let's say that they show you curtain number three. And so curtain number three has the goat. And then they'll ask you, do you want to switch to curtain number two? And the question here is, does it make a difference? Are you better off holding fast, sticking to your guns, staying with the original guess? Are you better off switching to whatever curtain is left? Or does it not matter? It's just random probability, and it's not going to make a difference whether you switch or not. So that is the brain teaser. Pause the video now. I encourage you to think about it. In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all. So now I've assumed that you've unpaused it. You've thought deeply about it. Perhaps you have an opinion on it. Now let's work it through a little bit. And at any point, I encourage you to pause it and kind of extrapolate beyond what I've already talked about. So let's think about the game show from the show's point of view. So the show knows where there's the goat and where there isn't the goat. So let's door number one, door number two, and door number three. So let's say that our prize is right over here. So our prize is the car. Our prize is the car, and that we have a goat over here, and over here we also have-- maybe we have two goats in this situation. So what are we going to do as the game show? Remember, the contestants don't know this. We know this. So the contestant picks door number one right over here. Then we can't lift door number two because there's a car back there. We're going to lift door number three, and we're going to expose this goat. In which case, it probably would be good for the person to switch. If the person picks door number two, then we as the game show can show either door number one or door number three, and then it actually does not make sense for the person to switch. If they picked door number three, then we have to show door number one because we can't pick door number two. And in that case, it actually makes a lot of sense for the person to switch. Now, with that out of the way, let's think about the probabilities given the two strategies. So if you don't switch, or another way to think about this strategy is you always stick to your guns. You always stick to your first guess. Well, in that situation, what is your probability of winning? Well, there's three doors. The prize is equally likely to be behind any one of them. So there's three possibilities. One has the outcome that you desire. The probability of winning will be 1/3 if you don't switch. Likewise, your probability of losing, well, there's two ways that you can lose out of three possibilities. It's going to be 2/3. And these are the only possibilities, and these add up to one right over here. So don't switch, 1/3 chance of winning. Now let's think about the switching situation. So let's say always-- when you always switch. Let's think about how this might play out. What is your probability of winning? And before we even think about that, think about how you would win if you always switch. So if you pick wrong the first time, they're going to show you this. And so you should always switch. So if you pick door number one, they're going to show you door number three. You should switch. If you picked wrong door number three, they're going to show you door number one. You should switch. So if you picked wrong and switch, you will always win. Let me write this down. And this insight actually came from one of the middle school students in the summer camp that Khan Academy was running. It's actually a fabulous way to think about this. So if you pick wrong, so step one, so initial pick is wrong, so you pick one of the two wrong doors, and then step two, you always switch, you will land on the car. Because if you picked one of the wrong doors, they're going to have to show the other wrong door. And so if you switch, you're going to end up on the right answer. So what is the probability of winning if you always switch? Well, it's going to be the probability that you initially picked wrong. Well, what's the probability that you initially picked wrong? Well, there's two out of the three ways to initially pick wrong. So you actually have a 2/3 chance of winning. There's a 2/3 chance you're going to pick wrong and then switch into the right one. Likewise, what's your probability of losing, given that you're always going to switch? Well, the way that you would lose is you pick right, you picked correctly. In step two, they're going to show one of the two empty or non-prize doors. And then step three, you're going to switch into the other empty door. But either way, you're definitely going to switch. So the only way to lose, if you're always going to switch, is to pick the right the first time. Well, what's the probability of you picking right the first time? Well, that is only 1/3. So you see here, it's sometimes counter-intuitive, but hopefully this makes sense as to why it isn't. You have a 1/3 chance of winning if you stick to your guns, and a 2/3 chance of winning if you always switch. Another way to think about it is, when you first make your initial pick, there's a 1/3 chance that it's there, and there's a 2/3 chance that it's in one of the other two doors. And they're going to empty out one of them. So when you switch, you essentially are capturing that 2/3 probability. And we see that right there. So hopefully you enjoyed that.