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Current time:0:00Total duration:7:23

Video transcript

let's now tackle a classic thought experiment in probability called the monty hall problem and it's called the monty hall problem because monty hall was the game show host and let's make a deal where they would set up a situation very similar to the monty 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 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 Monty Hall and let's make a deal crew we'll make it a little bit more interesting they won't just 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 will 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 will ask you do you want to switch 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 brainteaser start 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 what whether it makes any difference at all so now I've assumed that you've unpause 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 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 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 goat over here and over here we also have maybe we have two goats in this situation so what are we going to do is the gameshow 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 we're going to expose this goat in which case it probably would be good for the for the person to switch if the person picks if the person picks door number two then we as the gameshow 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 is 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 so let's if you don't switch 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 what is your probability of winning well there's three doors the the prize is equally likely to behind any one of them so there's three possibilities one has the outcome that you desire the probability of winning will be one-third if you don't switch likewise your probability of losing your probability of losing well there's two ways that you can lose out of three possibilities it's going to be two-thirds and these are the only possibilities and these add up to one right over here so don't switch one third chance of winning now let's think about the switching situation so let's say always when you always switch let's think let's think about how this might play out what is your probability of winning 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 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 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 if it's step one so initial pick is wrong initial pick is wrong so you pick one of the two wrong doors and then in step two you always switch you always 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 two-thirds chance of winning there's a two-thirds chance you're going to pick wrong and then switch into the right one likewise what's your probability of losing 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 pick right you pick correctly and step two they're going to show an empty 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 you are going to switch into the other 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 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 it here and it sometimes counterintuitive but hopefully this makes sense as to why it isn't you have a one-third chance of winning if you stick to your guns and a two-thirds chance of winning if you always switch another way to think about it is when you first make your initial pick there's a one-third chance that it's there there's a one-third chance there and there's a 2/3 chance there's a two-thirds 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