WEBVTT 00:00:00.058 --> 00:00:02.425 You have probably been told that you should switch, 00:00:02.508 --> 00:00:04.468 and you have probably not believed it. 00:00:04.551 --> 00:00:06.349 So let us not argue about it. 00:00:06.432 --> 00:00:09.983 Let us fix a choice, write down every place the car could be, 00:00:10.066 --> 00:00:12.377 and count what happens in each one. 00:00:13.460 --> 00:00:17.130 Here are the rules, so that we are arguing about the same game. 00:00:17.326 --> 00:00:18.627 Three doors. 00:00:18.824 --> 00:00:22.935 Behind one of them is a car, and behind each of the other two is a goat. 00:00:23.038 --> 00:00:25.512 You choose a door, and it stays shut. 00:00:26.309 --> 00:00:29.477 Then the host opens one of the two doors you did not choose, 00:00:29.560 --> 00:00:31.857 and what he shows you is always a goat. 00:00:31.940 --> 00:00:34.924 He can always do that, because he knows where the car is. 00:00:35.028 --> 00:00:38.338 Two doors are left shut, and he offers you the swap. 00:00:39.042 --> 00:00:42.746 Two things get fixed now, and then nothing else moves. 00:00:42.850 --> 00:00:44.232 First the doors. 00:00:44.336 --> 00:00:46.693 Number them one, two and three. 00:00:47.490 --> 00:00:49.232 Second, your choice. 00:00:49.545 --> 00:00:53.040 You pick door one, and it stays picked for the whole lecture. 00:00:53.353 --> 00:00:54.988 Nothing is lost by fixing it, 00:00:55.071 --> 00:00:58.845 because before you choose, the three doors are identical to you. 00:00:59.758 --> 00:01:02.139 So the only thing still free is the car. 00:01:02.336 --> 00:01:04.876 It is behind door one, or door two, 00:01:04.959 --> 00:01:08.675 or door three, and each of those is equally likely. 00:01:08.872 --> 00:01:12.658 That is three cases, and I am going to walk all three of them. 00:01:13.454 --> 00:01:15.043 And here is where this ends up, 00:01:15.126 --> 00:01:18.354 so that you can watch it arrive rather than take it from me. 00:01:18.667 --> 00:01:21.059 Staying wins one time in three. 00:01:21.163 --> 00:01:23.625 Switching wins two times in three. 00:01:24.329 --> 00:01:27.093 You are not asked to believe those two lines yet. 00:01:27.197 --> 00:01:29.415 They are the thing being tested. 00:01:29.612 --> 00:01:31.830 So: the first case. 00:01:32.626 --> 00:01:36.227 The car is behind door one, which is the door you picked. 00:01:36.423 --> 00:01:38.685 The other two doors have goats behind them. 00:01:38.768 --> 00:01:42.554 Remember that we can see all three of them, and the player cannot. 00:01:43.095 --> 00:01:47.891 The host has to pick a door that is not yours and does not have the car behind it. 00:01:48.088 --> 00:01:52.976 Here both of the doors he is allowed to touch have goats, so he has a free choice. 00:01:53.173 --> 00:01:55.252 Say he opens door three. 00:01:55.956 --> 00:01:57.846 Now count the two strategies. 00:01:57.929 --> 00:02:00.716 If you stay on door one, you win the car. 00:02:00.913 --> 00:02:03.782 If you switch to door two, you get a goat. 00:02:04.094 --> 00:02:07.880 This is the case where staying is right, and it is the only one. 00:02:08.677 --> 00:02:11.893 And his free choice here changes nothing about the count. 00:02:12.206 --> 00:02:17.060 Whichever of the two he opens, you are staying on a car, or switching to a goat. 00:02:17.256 --> 00:02:20.519 Case one is the case that costs you the car if you switch. 00:02:21.316 --> 00:02:22.582 Case two. 00:02:22.686 --> 00:02:27.818 The car is behind door two, so your own door has a goat, and so does door three. 00:02:28.870 --> 00:02:31.263 Now watch what the host is allowed to do. 00:02:31.366 --> 00:02:35.800 He cannot touch your door, and he cannot touch door two, because the car is there. 00:02:35.883 --> 00:02:38.273 So he has exactly one legal move left. 00:02:38.356 --> 00:02:40.249 He must open door three. 00:02:40.674 --> 00:02:42.065 He is not choosing there. 00:02:42.148 --> 00:02:46.291 He is being pushed, by the car, onto the one door he is still allowed to open. 00:02:46.374 --> 00:02:48.859 And that push is the whole of this problem. 00:02:49.656 --> 00:02:51.085 So count again. 00:02:51.281 --> 00:02:53.848 Stay on door one and you have a goat. 00:02:54.045 --> 00:02:56.774 Switch to door two and you have the car. 00:02:56.970 --> 00:02:58.980 Switching wins this case. 00:02:59.776 --> 00:03:03.213 Case three, and it is the mirror of case two. 00:03:03.410 --> 00:03:05.570 The car is behind door three. 00:03:05.767 --> 00:03:08.670 Your door is a goat, and so is door two. 00:03:09.095 --> 00:03:10.405 He cannot touch your door, 00:03:10.488 --> 00:03:13.667 and he cannot touch door three, because that is where the car is. 00:03:13.750 --> 00:03:18.918 So he is forced onto door two. Forced again, for the second time out of three. 00:03:19.970 --> 00:03:24.811 Stay, and you have the goat behind door one. Switch, and you have the car. 00:03:25.008 --> 00:03:26.669 Switching wins again. 00:03:27.582 --> 00:03:29.461 There is the whole enumeration. 00:03:29.544 --> 00:03:33.678 Three cases, equally likely, nothing left out and nothing hidden. 00:03:33.874 --> 00:03:35.593 Read the staying column: 00:03:35.790 --> 00:03:37.997 one win and two losses. 00:03:38.793 --> 00:03:40.744 Now read the switching column: 00:03:40.848 --> 00:03:42.915 two wins and one loss. 00:03:43.112 --> 00:03:45.583 One in three against two in three, 00:03:45.666 --> 00:03:47.954 and it did not come out of a formula. 00:03:48.058 --> 00:03:50.752 It came out of writing down every case. 00:03:51.456 --> 00:03:54.647 And if you want to know where the fifty fifty feeling comes from, 00:03:54.730 --> 00:03:58.922 it comes from looking at two shut doors and treating them as interchangeable. 00:03:59.235 --> 00:04:00.513 They are not. 00:04:00.965 --> 00:04:03.230 One of them you picked blind out of three. 00:04:03.426 --> 00:04:06.237 The other one is whatever survived the host. 00:04:07.033 --> 00:04:09.472 So here is the sentence that matters. 00:04:09.576 --> 00:04:12.814 In two of the three cases the host had no choice at all. 00:04:12.897 --> 00:04:17.913 He was forced, and which door he was forced onto depended on where the car was. 00:04:18.710 --> 00:04:21.668 That is information, and switching is how you collect it. 00:04:21.751 --> 00:04:24.571 Which tells you exactly which assumption is doing the work, 00:04:24.654 --> 00:04:26.486 and gives us something to test. 00:04:26.569 --> 00:04:28.590 What if the host is not forced? 00:04:29.712 --> 00:04:34.238 So change exactly one thing about the game. The host forgets where the car is. 00:04:34.321 --> 00:04:36.874 He still opens one of the two doors you did not pick, 00:04:36.957 --> 00:04:40.069 but now he chooses between them by flipping a coin. 00:04:40.866 --> 00:04:43.861 Which means that sometimes he opens the car himself. 00:04:44.058 --> 00:04:46.448 When that happens there is nothing left to decide, 00:04:46.531 --> 00:04:48.970 so we throw the round away and deal again. 00:04:49.282 --> 00:04:51.117 Everything else is untouched. 00:04:51.601 --> 00:04:54.318 Now count it again, the same way as before. 00:04:54.515 --> 00:04:57.462 Three places for the car, but each of those splits in two, 00:04:57.545 --> 00:05:00.367 because there are two doors his coin might send him to. 00:05:00.819 --> 00:05:02.283 Six branches. 00:05:03.795 --> 00:05:05.538 Same picture as before. 00:05:05.850 --> 00:05:06.814 Door one is yours, 00:05:06.814 --> 00:05:12.039 and the car is behind door one, door two or door three, exactly as it was. 00:05:12.836 --> 00:05:15.158 Start with the car behind door one. 00:05:15.471 --> 00:05:17.771 He flips, and opens door two: 00:05:17.967 --> 00:05:20.093 a goat, and staying wins. 00:05:20.405 --> 00:05:23.364 Or the coin lands the other way and he opens door three: 00:05:23.447 --> 00:05:26.165 also a goat, and staying wins again. 00:05:26.868 --> 00:05:29.168 Now the car is behind door two. 00:05:29.364 --> 00:05:31.232 If his coin sends him to door two, 00:05:31.315 --> 00:05:34.589 he has opened the car himself, and the round is void. 00:05:34.693 --> 00:05:39.361 If it sends him to door three, he shows a goat, and switching wins. 00:05:40.413 --> 00:05:43.490 And the car behind door three is the mirror of that. 00:05:43.687 --> 00:05:48.413 His coin sends him to door two, where he shows a goat, and switching wins. 00:05:48.609 --> 00:05:50.210 Or it sends him to door three, 00:05:50.293 --> 00:05:53.684 where he has shown the car, and the round is void again. 00:05:54.387 --> 00:05:59.427 Six branches, all equally likely, because the car is uniform and the coin is fair. 00:05:59.623 --> 00:06:02.397 Two of them end with the car standing there in the open, 00:06:02.480 --> 00:06:04.373 and those two get thrown out. 00:06:05.425 --> 00:06:07.339 So look at the four that survive. 00:06:07.422 --> 00:06:09.208 In two of them staying wins. 00:06:09.291 --> 00:06:11.428 In the other two switching wins. 00:06:11.625 --> 00:06:14.609 Two against two, and the advantage has gone. 00:06:15.406 --> 00:06:17.868 And look at what that does to your own door. 00:06:18.181 --> 00:06:21.175 In the first version it sat at one third whatever he did, 00:06:21.258 --> 00:06:23.813 because he could never fail to show a goat. 00:06:24.265 --> 00:06:27.595 Here, the fact that he happened to miss the car is itself evidence, 00:06:27.678 --> 00:06:30.628 and it lifts your door from one third to one half. 00:06:31.425 --> 00:06:32.935 Nothing else moved. 00:06:33.038 --> 00:06:35.034 Same doors, same first choice, 00:06:35.117 --> 00:06:39.076 same one in three that the car is behind door one at the start. 00:06:39.691 --> 00:06:43.755 The only thing that changed is whether the host knew where the car was. 00:06:44.668 --> 00:06:47.116 That is the real content of this puzzle, 00:06:47.199 --> 00:06:49.833 and it is why the first argument worked at all. 00:06:49.916 --> 00:06:54.433 A host who is forced two times in three is telling you something. 00:06:54.629 --> 00:06:57.881 A host who is flipping a coin is telling you nothing. 00:06:58.678 --> 00:07:01.567 So if the switching argument has always felt slippery, 00:07:01.650 --> 00:07:03.891 that is the joint it turns on. 00:07:04.088 --> 00:07:07.189 Now let me turn the numbers up until the thing is obvious. 00:07:08.238 --> 00:07:12.338 Last picture, and this is the one that usually finishes the argument. 00:07:12.534 --> 00:07:13.986 A hundred doors. 00:07:14.183 --> 00:07:16.575 One car, ninety nine goats. 00:07:16.772 --> 00:07:20.000 You pick the door in the corner there, and it stays picked. 00:07:20.704 --> 00:07:24.663 The chance that you found the car with that pick is one in a hundred. 00:07:24.860 --> 00:07:26.416 You will not argue with me about that, 00:07:26.416 --> 00:07:30.016 because there was nothing at all to go on when you chose. 00:07:30.928 --> 00:07:34.899 Now the host, who knows exactly where the car is, opens ninety eight 00:07:34.899 --> 00:07:38.522 of the other doors, and every single one of them has a goat behind it. 00:07:38.719 --> 00:07:40.099 He is not lucky. 00:07:40.182 --> 00:07:43.526 He is stepping around the car, ninety eight times in a row. 00:07:44.439 --> 00:07:45.937 Two doors are left. 00:07:46.041 --> 00:07:50.964 Yours, and the one door out of ninety nine that he walked past every single time. 00:07:51.761 --> 00:07:54.096 So ask the question in this form. 00:07:54.292 --> 00:07:57.033 Did your one in a hundred just turn into one in two? 00:07:57.230 --> 00:07:58.598 Of course it did not. 00:07:58.681 --> 00:08:01.979 Your door is still the door you picked blind out of a hundred. 00:08:02.775 --> 00:08:06.872 All the probability that was spread thinly over the other ninety nine has been swept, 00:08:06.955 --> 00:08:09.427 by a host who was never allowed to touch the car, 00:08:09.510 --> 00:08:11.960 onto the single door he left standing. 00:08:12.757 --> 00:08:16.435 And the three door game is that same game with the numbers turned down. 00:08:16.518 --> 00:08:20.313 Your door holds a third. The other two hold two thirds between them, 00:08:20.396 --> 00:08:24.506 and the host pours all of it onto whichever door he does not open. 00:08:25.303 --> 00:08:27.588 So here is the whole thing on one board. 00:08:27.671 --> 00:08:30.398 If the host knows where the car is and has to avoid it, 00:08:30.481 --> 00:08:32.931 switching wins two times in three. 00:08:33.728 --> 00:08:37.198 If he is opening doors at random and merely happened to miss the car, 00:08:37.281 --> 00:08:41.136 it is one half either way, and you may do whatever you like. 00:08:41.561 --> 00:08:43.034 Three things to keep. 00:08:43.117 --> 00:08:46.575 Your first door keeps whatever chance it had at the moment you picked it. 00:08:46.658 --> 00:08:50.708 A host who must avoid the car sweeps all the rest onto a single door. 00:08:50.791 --> 00:08:53.799 And a host who is guessing sweeps nothing at all. 00:08:54.711 --> 00:08:57.868 Which is why this puzzle was never really about the doors. 00:08:57.951 --> 00:09:01.144 It was about the host, and whether he was filtering. 00:09:01.248 --> 00:09:03.222 When he is, switch.