{"version":1,"lectureId":"01M14V06HJH7CGGZK9BX32DPD5","attempt":0,"publication":{"slug":"monty-hall","title":"The Monty Hall Problem: Why Switching Wins","subject":"statistics","summary":"A case by case proof, built for someone who does not believe it. One door is fixed, every place the car could be is written down, and the host's forced move is shown in each case together with the outcome of staying and of switching. Then the host's knowledge is put on trial: he forgets where the car is, opens a door by coin flip, and sometimes reveals the car himself, and the two thirds advantage collapses to a coin flip. A hundred doors close the argument.","metaDescription":"Fix one door, list every place the car can be, and count. Then watch the advantage vanish when the host stops knowing where the car is.","transcript":"You have probably been told that you should switch, and you have probably not believed it. So let us not argue about it. Let us fix a choice, write down every place the car could be, and count what happens in each one. Here are the rules, so that we are arguing about the same game. Three doors. Behind one of them is a car, and behind each of the other two is a goat. You choose a door, and it stays shut. Then the host opens one of the two doors you did not choose, and what he shows you is always a goat. He can always do that, because he knows where the car is. Two doors are left shut, and he offers you the swap. Two things get fixed now, and then nothing else moves. First the doors. Number them one, two and three. Second, your choice. You pick door one, and it stays picked for the whole lecture. Nothing is lost by fixing it, because before you choose, the three doors are identical to you. So the only thing still free is the car. It is behind door one, or door two, or door three, and each of those is equally likely. That is three cases, and I am going to walk all three of them. And here is where this ends up, so that you can watch it arrive rather than take it from me. Staying wins one time in three. Switching wins two times in three. You are not asked to believe those two lines yet. They are the thing being tested. So: the first case. The car is behind door one, which is the door you picked. The other two doors have goats behind them. Remember that we can see all three of them, and the player cannot. The host has to pick a door that is not yours and does not have the car behind it. Here both of the doors he is allowed to touch have goats, so he has a free choice. Say he opens door three. Now count the two strategies. If you stay on door one, you win the car. If you switch to door two, you get a goat. This is the case where staying is right, and it is the only one. And his free choice here changes nothing about the count. Whichever of the two he opens, you are staying on a car, or switching to a goat. Case one is the case that costs you the car if you switch. Case two. The car is behind door two, so your own door has a goat, and so does door three. Now watch what the host is allowed to do. He cannot touch your door, and he cannot touch door two, because the car is there. So he has exactly one legal move left. He must open door three. He is not choosing there. He is being pushed, by the car, onto the one door he is still allowed to open. And that push is the whole of this problem. So count again. Stay on door one and you have a goat. Switch to door two and you have the car. Switching wins this case. Case three, and it is the mirror of case two. The car is behind door three. Your door is a goat, and so is door two. He cannot touch your door, and he cannot touch door three, because that is where the car is. So he is forced onto door two. Forced again, for the second time out of three. Stay, and you have the goat behind door one. Switch, and you have the car. Switching wins again. There is the whole enumeration. Three cases, equally likely, nothing left out and nothing hidden. Read the staying column: one win and two losses. Now read the switching column: two wins and one loss. One in three against two in three, and it did not come out of a formula. It came out of writing down every case. And if you want to know where the fifty fifty feeling comes from, it comes from looking at two shut doors and treating them as interchangeable. They are not. One of them you picked blind out of three. The other one is whatever survived the host. So here is the sentence that matters. In two of the three cases the host had no choice at all. He was forced, and which door he was forced onto depended on where the car was. That is information, and switching is how you collect it. Which tells you exactly which assumption is doing the work, and gives us something to test. What if the host is not forced? So change exactly one thing about the game. The host forgets where the car is. He still opens one of the two doors you did not pick, but now he chooses between them by flipping a coin. Which means that sometimes he opens the car himself. When that happens there is nothing left to decide, so we throw the round away and deal again. Everything else is untouched. Now count it again, the same way as before. Three places for the car, but each of those splits in two, because there are two doors his coin might send him to. Six branches. Same picture as before. Door one is yours, and the car is behind door one, door two or door three, exactly as it was. Start with the car behind door one. He flips, and opens door two: a goat, and staying wins. Or the coin lands the other way and he opens door three: also a goat, and staying wins again. Now the car is behind door two. If his coin sends him to door two, he has opened the car himself, and the round is void. If it sends him to door three, he shows a goat, and switching wins. And the car behind door three is the mirror of that. His coin sends him to door two, where he shows a goat, and switching wins. Or it sends him to door three, where he has shown the car, and the round is void again. Six branches, all equally likely, because the car is uniform and the coin is fair. Two of them end with the car standing there in the open, and those two get thrown out. So look at the four that survive. In two of them staying wins. In the other two switching wins. Two against two, and the advantage has gone. And look at what that does to your own door. In the first version it sat at one third whatever he did, because he could never fail to show a goat. Here, the fact that he happened to miss the car is itself evidence, and it lifts your door from one third to one half. Nothing else moved. Same doors, same first choice, same one in three that the car is behind door one at the start. The only thing that changed is whether the host knew where the car was. That is the real content of this puzzle, and it is why the first argument worked at all. A host who is forced two times in three is telling you something. A host who is flipping a coin is telling you nothing. So if the switching argument has always felt slippery, that is the joint it turns on. Now let me turn the numbers up until the thing is obvious. Last picture, and this is the one that usually finishes the argument. A hundred doors. One car, ninety nine goats. You pick the door in the corner there, and it stays picked. The chance that you found the car with that pick is one in a hundred. You will not argue with me about that, because there was nothing at all to go on when you chose. Now the host, who knows exactly where the car is, opens ninety eight of the other doors, and every single one of them has a goat behind it. He is not lucky. He is stepping around the car, ninety eight times in a row. Two doors are left. Yours, and the one door out of ninety nine that he walked past every single time. So ask the question in this form. Did your one in a hundred just turn into one in two? Of course it did not. Your door is still the door you picked blind out of a hundred. All the probability that was spread thinly over the other ninety nine has been swept, by a host who was never allowed to touch the car, onto the single door he left standing. And the three door game is that same game with the numbers turned down. Your door holds a third. The other two hold two thirds between them, and the host pours all of it onto whichever door he does not open. So here is the whole thing on one board. If the host knows where the car is and has to avoid it, switching wins two times in three. If he is opening doors at random and merely happened to miss the car, it is one half either way, and you may do whatever you like. Three things to keep. Your first door keeps whatever chance it had at the moment you picked it. A host who must avoid the car sweeps all the rest onto a single door. And a host who is guessing sweeps nothing at all. Which is why this puzzle was never really about the doors. It was about the host, and whether he was filtering. When he is, switch.","watch":{"version":1,"scenes":[{"title":"Fix Your Door, Then Count","start":0,"end":276.5765625,"objects":{"car_1":"a Math [green] that says \"$upright(\"car\")$\" drawn in stage","car_2":"a Math [green] that says \"$upright(\"car\")$\" drawn in stage","car_3":"a Math [green] that says \"$upright(\"car\")$\" drawn in stage","card":"a Title that says \"Introductory Probability — The Monty Hall Problem: Why Switching Wins\"","goat_1":"a Math [gray] that says \"$upright(\"goat\")$\" drawn in stage","goat_2":"a Math [gray] that says \"$upright(\"goat\")$\" drawn in stage","goat_3":"a Math [gray] that says \"$upright(\"goat\")$\" drawn in stage","num_1":"a Math [text] that says \"$1$\" drawn in stage","num_2":"a Math [text] that says \"$2$\" drawn in stage","num_3":"a Math [text] that says \"$3$\" drawn in stage","open_2":"a Math [red] that says \"$upright(\"opened\")$\" drawn in stage","open_3":"a Math [red] that says \"$upright(\"opened\")$\" drawn in stage","p_stay":"a Math [text] that says \"$P(upright(\"stay wins\")) = frac(1, 3)$\"","p_switch":"a Math [text] that says \"$P(upright(\"switch wins\")) = frac(2, 3)$\"","pick":"a Polygon [yellow] drawn in stage (vertices=((0.39999999999999997, 0.8), (3.0000000000000004, 0.8), (3.0000…, filled=False)","pick_tag":"a Math [yellow] that says \"$upright(\"your pick\")$\" drawn in stage","point":"a Point [yellow] drawn in stage (location=(1.7000000000000002, 2.6))","point_2":"a Point [yellow] drawn in stage (location=(5.1, 2.6))","point_3":"a Point [yellow] drawn in stage (location=(8.5, 2.6))","point_4":"a Point [yellow] drawn in stage (location=(1.7000000000000002, 2.6))","point_5":"a Point [yellow] drawn in stage (location=(8.5, 2.6))","polygon":"a Polygon [blue] drawn in stage (vertices=((0.6, 1.0), (2.8000000000000003, 1.0), (2.8000000000000003, 4.…, fill_opacity=0.18)","polygon_2":"a Polygon [blue] drawn in stage (vertices=((4.0, 1.0), (6.2, 1.0), (6.2, 4.2), (4.0, 4.2)), fill_opacity=0.18)","polygon_3":"a Polygon [blue] drawn in stage (vertices=((7.4, 1.0), (9.600000000000001, 1.0), (9.600000000000001, 4.2)…, fill_opacity=0.18)","question":"a Panel that says \"Three doors: one car, two goats. You pick a door. The host, who knows where the car is, opens one of the other two doors and shows you a goat. Then he offers you the swap. Stay, or switch?\"","stage":"a Figure (x_range=(0.0, 10.2), y_range=(0.0, 5.2), aspect=(10.2, 5.2))","tally":"a Table [text] that says \"Car behind Host opens Stay Switch 1 2 or 3 win lose 2 3 lose win 3 2 lose win\" (rows=(('Car behind', 'Host opens', 'Stay', 'Switch'), ('1', '2 or 3'…, header=True)"},"beats":[{"start":0,"say":"You have probably been told that you should switch, and you have probably not believed it. So let us not argue about it. Let us fix a choice, write down every place the car could be, and count what happens in each one.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[13.061,"card is hidden from the screen — left the board."]]},{"start":14.261,"say":"Here are the rules, so that we are arguing about the same game. Three doors. Behind one of them is a car, and behind each of the other two is a goat. You choose a door, and it stays shut.","live":null,"does":[[15.270999999999999,"question is shown on the screen, written out."]]},{"start":27.6325,"say":"Then the host opens one of the two doors you did not choose, and what he shows you is always a goat. He can always do that, because he knows where the car is. Two doors are left shut, and he offers you the swap.","live":["question"],"does":[[35.22500000000001,"question (the \"who knows where the car is\" part) is emphasized."],[40.054500000000004,"question moves to a new place on the board."]]},{"start":40.6545,"say":"Two things get fixed now, and then nothing else moves. First the doors. Number them one, two and three.","live":null,"does":[[40.6545,"stage is shown on the screen, written out."],[40.6545,"question (the \"who knows where the car is\" part) is no longer emphasized."],[44.997,"polygon is shown on the screen, written out."],[45.117,"num_1 is shown on the screen, written out."],[45.237,"polygon_2 is shown on the screen, written out."],[45.357,"num_2 is shown on the screen, written out."],[45.477,"polygon_3 is shown on the screen, written out."],[45.597,"num_3 is shown on the screen, written out."]]},{"start":49.346999999999994,"say":"Second, your choice. You pick door one, and it stays picked for the whole lecture. Nothing is lost by fixing it, because before you choose, the three doors are identical to you.","live":["question","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3"],"does":[[52.34199999999999,"pick is shown on the screen, written out."],[52.34199999999999,"pick_tag is shown on the screen, written out."]]},{"start":61.0115,"say":"So the only thing still free is the car. It is behind door one, or door two, or door three, and each of those is equally likely. That is three cases, and I am going to walk all three of them.","live":["question","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag"],"does":[[64.994,"point is shown on the screen, grown."],[65.79500000000002,"point_2 is shown on the screen, grown."],[66.596,"point_3 is shown on the screen, grown."],[66.994,"point is hidden from the screen."],[67.79500000000002,"point_2 is hidden from the screen."],[68.596,"point_3 is hidden from the screen."]]},{"start":72.9605,"say":"And here is where this ends up, so that you can watch it arrive rather than take it from me. Staying wins one time in three. Switching wins two times in three.","live":null,"does":[[77.761,"stage moves to a new place on the board."],[77.761,"p_stay is shown on the screen, written out."],[80.141,"p_switch is shown on the screen, written out."]]},{"start":82.8365,"say":"You are not asked to believe those two lines yet. They are the thing being tested. So: the first case.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag"],"does":[[89.25099999999999,"tally is shown on the screen, written out."]]},{"start":90.6985,"say":"The car is behind door one, which is the door you picked. The other two doors have goats behind them. Remember that we can see all three of them, and the player cannot.","live":null,"does":[[91.20899999999999,"car_1 is shown on the screen, written out."],[96.20199999999998,"goat_2 is shown on the screen, written out."],[96.40199999999999,"goat_3 is shown on the screen, written out."]]},{"start":102.375,"say":"The host has to pick a door that is not yours and does not have the car behind it. Here both of the doors he is allowed to touch have goats, so he has a free choice. Say he opens door three.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","car_1","goat_2","goat_3"],"does":[[113.22999999999999,"open_3 is shown on the screen, written out."]]},{"start":115.4435,"say":"Now count the two strategies. If you stay on door one, you win the car. If you switch to door two, you get a goat. This is the case where staying is right, and it is the only one.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","car_1","goat_2","goat_3","open_3"],"does":[[116.047,"tally is shown on the screen, written out."],[119.49499999999999,"tally (the \"win\" part) is emphasized."],[121.27199999999999,"tally (the \"win\" part) is no longer emphasized."],[123.036,"tally (the \"lose\" part) is emphasized."]]},{"start":129.035,"say":"And his free choice here changes nothing about the count. Whichever of the two he opens, you are staying on a car, or switching to a goat. Case one is the case that costs you the car if you switch.","live":null,"does":[[133.122,"tally is indicated — a transient flash."]]},{"start":142.418,"say":"Case two. The car is behind door two, so your own door has a goat, and so does door three.","live":null,"does":[[142.418,"car_1 is hidden from the screen."],[142.418,"goat_2 is hidden from the screen."],[142.418,"open_3 is hidden from the screen."],[142.418,"tally (the \"lose\" part) is no longer emphasized."],[144.415,"car_2 is shown on the screen, written out."],[147.073,"goat_1 is shown on the screen, written out."]]},{"start":150.03050000000002,"say":"Now watch what the host is allowed to do. He cannot touch your door, and he cannot touch door two, because the car is there. So he has exactly one legal move left. He must open door three.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_3","car_2","goat_1"],"does":[[161.14,"open_3 is shown on the screen, written out."]]},{"start":163.192,"say":"He is not choosing there. He is being pushed, by the car, onto the one door he is still allowed to open. And that push is the whole of this problem.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_3","open_3","car_2","goat_1"],"does":[[165.665,"open_3 is indicated — a transient flash."],[166.315,"car_2 is indicated — a transient flash."]]},{"start":172.42950000000002,"say":"So count again. Stay on door one and you have a goat. Switch to door two and you have the car. Switching wins this case.","live":null,"does":[[173.02100000000002,"tally is shown on the screen, written out."],[180.71900000000002,"tally (the \"win\" part) is emphasized."]]},{"start":183.13000000000002,"say":"Case three, and it is the mirror of case two. The car is behind door three. Your door is a goat, and so is door two.","live":null,"does":[[183.13000000000002,"open_3 is hidden from the screen."],[183.13000000000002,"car_2 is hidden from the screen."],[183.13000000000002,"goat_3 is hidden from the screen."],[183.13000000000002,"tally (the \"win\" part) is no longer emphasized."],[187.46,"car_3 is shown on the screen, written out."],[190.44400000000002,"goat_2 is shown on the screen, written out."]]},{"start":193.44750000000002,"say":"He cannot touch your door, and he cannot touch door three, because that is where the car is. So he is forced onto door two. Forced again, for the second time out of three.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_2","goat_1","car_3"],"does":[[199.949,"open_2 is shown on the screen, written out."]]},{"start":205.56500000000003,"say":"Stay, and you have the goat behind door one. Switch, and you have the car. Switching wins again.","live":["question","p_stay","p_switch","stage","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_2","goat_1","car_3","open_2"],"does":[[206.04100000000003,"tally is shown on the screen, written out."],[212.728,"tally (the \"win\" part) is emphasized."]]},{"start":213.98350000000002,"say":"There is the whole enumeration. Three cases, equally likely, nothing left out and nothing hidden. Read the staying column: one win and two losses.","live":null,"does":[[213.98350000000002,"tally (the \"win\" part) is no longer emphasized."],[221.38500000000002,"tally (the \"column=3\" part) is emphasized."]]},{"start":225.09650000000002,"say":"Now read the switching column: two wins and one loss. One in three against two in three, and it did not come out of a formula. It came out of writing down every case.","live":null,"does":[[225.09650000000002,"tally (the \"column=3\" part) is no longer emphasized."],[226.037,"tally (the \"column=4\" part) is emphasized."],[228.615,"p_stay (the \"frac(1, 3)\" part) is emphasized."],[230.867,"p_switch (the \"frac(2, 3)\" part) is emphasized."]]},{"start":237.58550000000002,"say":"And if you want to know where the fifty fifty feeling comes from, it comes from looking at two shut doors and treating them as interchangeable. They are not. One of them you picked blind out of three. The other one is whatever survived the host.","live":null,"does":[[237.58550000000002,"tally (the \"column=4\" part) is no longer emphasized."],[237.58550000000002,"p_stay (the \"frac(1, 3)\" part) is no longer emphasized."],[237.58550000000002,"p_switch (the \"frac(2, 3)\" part) is no longer emphasized."],[246.792,"point_4 is shown on the screen, grown."],[248.792,"point_4 is hidden from the screen."],[249.567,"point_5 is shown on the screen, grown."],[251.567,"point_5 is hidden from the screen."]]},{"start":251.68750000000003,"say":"So here is the sentence that matters. In two of the three cases the host had no choice at all. He was forced, and which door he was forced onto depended on where the car was.","live":null,"does":[[258.514,"tally is indicated — a transient flash."],[258.714,"tally is indicated — a transient flash."]]},{"start":263.6075,"say":"That is information, and switching is how you collect it. 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Six branches.","live":["forgetful","unchanged","heading"],"does":[[301.0075625,"forgetful (the \"sometimes he opens the car\" part) is no longer emphasized."],[308.13606250000004,"forgetful is hidden from the screen — left the board."],[308.13606250000004,"heading is hidden from the screen — left the board."],[308.13606250000004,"unchanged is hidden from the screen — left the board."]]},{"start":309.3360625,"say":"Same picture as before. Door one is yours, and the car is behind door one, door two or door three, exactly as it was.","live":[],"does":[[309.3360625,"heading_count is shown on the screen, written out."],[309.3360625,"stage is shown on the screen, written out."],[309.9515625,"polygon is shown on the screen, written out."],[310.0715625,"num_1 is shown on the screen, written out."],[310.19156250000003,"polygon_2 is shown on the screen, written out."],[310.31156250000004,"num_2 is shown on the screen, written out."],[310.43156250000004,"polygon_3 is shown on the screen, written out."],[310.55156250000005,"num_3 is shown on the screen, written out."],[312.09956250000005,"pick is shown on the screen, written out."],[312.09956250000005,"pick_tag is shown on the screen, written out."],[315.74456250000003,"stage moves to a new place on the board."],[315.74456250000003,"branches is shown on the screen, written out."]]},{"start":318.0630625,"say":"Start with the car behind door one. He flips, and opens door two: a goat, and staying wins. Or the coin lands the other way and he opens door three: also a goat, and staying wins again.","live":["stage","heading_count","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag"],"does":[[318.4695625,"car_1 is shown on the screen, written out."],[318.4695625,"goat_2 is shown on the screen, written out."],[318.4695625,"goat_3 is shown on the screen, written out."],[322.39356250000003,"open_2 is shown on the screen, written out."],[323.26356250000003,"branches is shown on the screen, written out."],[325.5045625,"open_2 is hidden from the screen."],[327.6295625,"open_3 is shown on the screen, written out."],[328.1755625,"branches is shown on the screen, written out."]]},{"start":331.5960625,"say":"Now the car is behind door two. If his coin sends him to door two, he has opened the car himself, and the round is void. If it sends him to door three, he shows a goat, and switching wins.","live":["stage","heading_count","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","car_1","goat_2","goat_3","open_3"],"does":[[331.5960625,"open_3 is hidden from the screen."],[331.5960625,"car_1 is hidden from the screen."],[331.5960625,"goat_2 is hidden from the screen."],[332.3395625,"car_2 is shown on the screen, written out."],[332.3395625,"goat_1 is shown on the screen, written out."],[335.09056250000003,"open_2 is shown on the screen, written out."],[336.68156250000004,"car_2 is indicated — a transient flash."],[338.37656250000003,"branches is shown on the screen, written out."],[339.2245625,"open_2 is hidden from the screen."],[339.4795625,"open_3 is shown on the screen, written out."],[341.8485625,"branches is shown on the screen, written out."]]},{"start":343.60906250000005,"say":"And the car behind door three is the mirror of that. His coin sends him to door two, where he shows a goat, and switching wins. Or it sends him to door three, where he has shown the car, and the round is void again.","live":["stage","heading_count","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_3","open_3","car_2","goat_1"],"does":[[343.60906250000005,"open_3 is hidden from the screen."],[343.60906250000005,"car_2 is hidden from the screen."],[343.60906250000005,"goat_3 is hidden from the screen."],[344.1545625,"car_3 is shown on the screen, written out."],[344.1545625,"goat_2 is shown on the screen, written out."],[347.6375625,"open_2 is shown on the screen, written out."],[350.4595625,"branches is shown on the screen, written out."],[352.10756250000003,"open_2 is hidden from the screen."],[353.1295625,"open_3 is shown on the screen, written out."],[354.18556250000006,"car_3 is indicated — a transient flash."],[355.9735625,"branches is shown on the screen, written out."]]},{"start":357.7115625,"say":"Six branches, all equally likely, because the car is uniform and the coin is fair. Two of them end with the car standing there in the open, and those two get thrown out.","live":["stage","heading_count","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_2","open_3","goat_1","car_3"],"does":[[363.62156250000004,"branches is indicated — a transient flash."],[363.8215625,"branches is indicated — a transient flash."]]},{"start":368.3490625,"say":"So look at the four that survive. In two of them staying wins. In the other two switching wins. Two against two, and the advantage has gone.","live":null,"does":[[371.5825625,"p_stay is shown on the screen, written out."],[373.6605625,"p_switch is shown on the screen, written out."]]},{"start":378.60256250000003,"say":"And look at what that does to your own door. In the first version it sat at one third whatever he did, because he could never fail to show a goat. Here, the fact that he happened to miss the car is itself evidence, and it lifts your door from one third to one half.","live":["p_stay","p_switch","stage","heading_count","polygon","num_1","polygon_2","num_2","polygon_3","num_3","pick","pick_tag","goat_2","open_3","goat_1","car_3"],"does":[[380.2505625,"pick is indicated — a transient flash."],[392.5455625,"p_stay (the \"frac(1, 2)\" part) is emphasized."]]},{"start":393.91206250000005,"say":"Nothing else moved. Same doors, same first choice, same one in three that the car is behind door one at the start. The only thing that changed is whether the host knew where the car was.","live":null,"does":[[394.8175625,"p_stay (the \"frac(1, 2)\" part) is no longer emphasized."],[403.6875625,"p_switch is indicated — a transient flash."]]},{"start":405.98256250000003,"say":"That is the real content of this puzzle, and it is why the first argument worked at all. A host who is forced two times in three is telling you something. A host who is flipping a coin is telling you nothing.","live":null,"does":[[413.9235625,"branches (the \"column=3\" part) is emphasized."]]},{"start":419.19056250000006,"say":"So if the switching argument has always felt slippery, that is the joint it turns on. Now let me turn the numbers up until the thing is obvious.","live":null,"does":[[419.19056250000006,"branches (the \"column=3\" part) is no longer emphasized."],[423.3465625,"p_switch is indicated — a transient flash."],[428.0126875,"branches is hidden from the screen — left the board."],[428.0126875,"heading_count is hidden from the screen — left the board."],[428.0126875,"p_stay is hidden from the screen — left the board."],[428.0126875,"p_switch is hidden from the screen — left the board."],[428.0126875,"stage is hidden from the screen — left the board."],[428.0126875,"polygon is hidden from the screen — stage left the board."],[428.0126875,"num_1 is hidden from the screen — stage left the board."],[428.0126875,"polygon_2 is hidden from the screen — stage left the board."],[428.0126875,"num_2 is hidden from the screen — stage left the board."],[428.0126875,"polygon_3 is hidden from the screen — stage left the board."],[428.0126875,"num_3 is hidden from the screen — stage left the board."],[428.0126875,"pick is hidden from the screen — stage left the board."],[428.0126875,"pick_tag is hidden from the screen — stage left the board."],[428.0126875,"goat_2 is hidden from the screen — stage left the board."],[428.0126875,"open_3 is hidden from the screen — stage left the board."],[428.0126875,"goat_1 is hidden from the screen — stage left the board."],[428.0126875,"car_3 is hidden from the screen — stage left the board."]]}]},{"title":"A Hundred Doors","start":429.0543541666667,"end":542.9256041666667,"objects":{"final_guesses":"a Math [text] that says \"$upright(\"host guesses: \") quad P(upright(\"switch wins\")) = frac(1, 2)$\"","final_knows":"a Math [text] that says \"$upright(\"host knows: \") quad P(upright(\"switch wins\")) = frac(2, 3)$\"","grid":"a Figure (x_range=(0.0, 21.0), y_range=(0.0, 10.0), aspect=(21.0, 10.0))","head_close":"a Heading that says \"What To Take Away\"","head_grid":"a Heading that says \"A Hundred Doors\"","p_one":"a Math [text] that 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Yours, and the one door out of ninety nine that he walked past every single time.","live":["grid","p_one","head_grid","polygon","polygon_73","polygon_101"],"does":[[463.4673541666667,"tag_mine is shown on the screen, written out."],[464.6743541666667,"polygon_102 is shown on the screen, written out."],[464.6743541666667,"tag_left is shown on the screen, written out."]]},{"start":468.8038541666667,"say":"So ask the question in this form. Did your one in a hundred just turn into one in two? Of course it did not. Your door is still the door you picked blind out of a hundred.","live":["grid","p_one","head_grid","polygon","polygon_73","polygon_101","tag_mine","polygon_102","tag_left"],"does":[[475.4683541666667,"p_one is indicated — a transient flash."]]},{"start":480.2713541666667,"say":"All the probability that was spread thinly over the other ninety nine has been swept, by a host who was never allowed to touch the car, onto the single door he left standing.","live":null,"does":[[484.0093541666667,"p_rest is shown on the screen, written out."],[488.8863541666667,"A box is drawn around p_rest."]]},{"start":490.3918541666667,"say":"And the three door game is that same game with the numbers turned down. Your door holds a third. The other two hold two thirds between them, and the host pours all of it onto whichever door he does not open.","live":["grid","p_one","p_rest","head_grid","polygon","polygon_73","polygon_101","tag_mine","polygon_102","tag_left"],"does":[[499.72635416666674,"polygon_102 is indicated — a transient flash."],[502.8143541666667,"grid is hidden from the screen — left the board."],[502.8143541666667,"polygon is hidden from the screen — grid left the board."],[502.8143541666667,"polygon_73 is hidden from the screen — grid left the board."],[502.8143541666667,"polygon_101 is hidden from the screen — grid left the board."],[502.8143541666667,"tag_mine is hidden from the screen — grid left the board."],[502.8143541666667,"polygon_102 is hidden from the screen — grid left the board."],[502.8143541666667,"tag_left is hidden from the screen — grid left the board."],[502.8143541666667,"head_grid is hidden from the screen — left the board."],[502.8143541666667,"p_one is hidden from the screen — left the board."],[502.8143541666667,"p_rest is hidden from the screen — left the board."]]},{"start":503.4143541666667,"say":"So here is the whole thing on one board. If the host knows where the car is and has to avoid it, switching wins two times in three.","live":[],"does":[[503.4143541666667,"head_close is shown on the screen, written out."],[506.88535416666673,"final_knows is shown on the screen, written out."]]},{"start":511.90935416666673,"say":"If he is opening doors at random and merely happened to miss the car, it is one half either way, and you may do whatever you like.","live":["final_knows","head_close"],"does":[[513.6153541666667,"final_guesses is shown on the screen, written out."]]},{"start":519.7538541666668,"say":"Three things to keep. Your first door keeps whatever chance it had at the moment you picked it. A host who must avoid the car sweeps all the rest onto a single door. And a host who is guessing sweeps nothing at all.","live":["final_knows","final_guesses","head_close"],"does":[[519.9743541666667,"takeaway is shown on the screen, written out."],[524.7343541666667,"takeaway (the \"Your first door keeps whatever chance it had when you picked it.\" part) is emphasized."],[527.0333541666668,"takeaway (the \"A host who must avoid the car sweeps all the rest onto one door.\" part) is emphasized."],[527.0333541666668,"takeaway (the \"Your first door keeps whatever chance it had when you picked it.\" part) is no longer emphasized."],[530.9113541666667,"takeaway (the \"A host who is guessing sweeps nothing, and it is a coin flip.\" part) is emphasized."],[530.9113541666667,"takeaway (the \"A host who must avoid the car sweeps all the rest onto one door.\" part) is no longer emphasized."]]},{"start":533.3918541666667,"say":"Which is why this puzzle was never really about the doors. It was about the host, and whether he was filtering. When he is, switch.","live":["final_knows","final_guesses","takeaway","head_close"],"does":[[533.3918541666667,"takeaway (the \"A host who is guessing sweeps nothing, and it is a coin flip.\" part) is no longer emphasized."],[540.8913541666667,"final_knows is indicated — a transient flash."],[541.8839375,"final_guesses is hidden from the screen — left the board."],[541.8839375,"final_knows is hidden from the screen — left the board."],[541.8839375,"head_close is hidden from the screen — left the board."],[541.8839375,"takeaway is hidden from the screen — left the board."]]}]}]},"durationSeconds":543,"chapters":[{"title":"Fix Your Door, Then Count","startSeconds":0,"narration":"You have probably been told that you should switch, and you have probably not believed it. So let us not argue about it. Let us fix a choice, write down every place the car could be, and count what happens in each one. Here are the rules, so that we are arguing about the same game. Three doors. Behind one of them is a car, and behind each of the other two is a goat. You choose a door, and it stays shut. Then the host opens one of the two doors you did not choose, and what he shows you is always a goat. He can always do that, because he knows where the car is. Two doors are left shut, and he offers you the swap. Two things get fixed now, and then nothing else moves. First the doors. Number them one, two and three. Second, your choice. You pick door one, and it stays picked for the whole lecture. Nothing is lost by fixing it, because before you choose, the three doors are identical to you. So the only thing still free is the car. It is behind door one, or door two, or door three, and each of those is equally likely. That is three cases, and I am going to walk all three of them. And here is where this ends up, so that you can watch it arrive rather than take it from me. Staying wins one time in three. Switching wins two times in three. You are not asked to believe those two lines yet. They are the thing being tested. So: the first case. The car is behind door one, which is the door you picked. The other two doors have goats behind them. Remember that we can see all three of them, and the player cannot. The host has to pick a door that is not yours and does not have the car behind it. Here both of the doors he is allowed to touch have goats, so he has a free choice. Say he opens door three. Now count the two strategies. If you stay on door one, you win the car. If you switch to door two, you get a goat. This is the case where staying is right, and it is the only one. And his free choice here changes nothing about the count. Whichever of the two he opens, you are staying on a car, or switching to a goat. Case one is the case that costs you the car if you switch. Case two. The car is behind door two, so your own door has a goat, and so does door three. Now watch what the host is allowed to do. He cannot touch your door, and he cannot touch door two, because the car is there. So he has exactly one legal move left. He must open door three. He is not choosing there. He is being pushed, by the car, onto the one door he is still allowed to open. And that push is the whole of this problem. So count again. Stay on door one and you have a goat. Switch to door two and you have the car. Switching wins this case. Case three, and it is the mirror of case two. The car is behind door three. Your door is a goat, and so is door two. He cannot touch your door, and he cannot touch door three, because that is where the car is. So he is forced onto door two. Forced again, for the second time out of three. Stay, and you have the goat behind door one. Switch, and you have the car. Switching wins again. There is the whole enumeration. Three cases, equally likely, nothing left out and nothing hidden. Read the staying column: one win and two losses. Now read the switching column: two wins and one loss. One in three against two in three, and it did not come out of a formula. It came out of writing down every case. And if you want to know where the fifty fifty feeling comes from, it comes from looking at two shut doors and treating them as interchangeable. They are not. One of them you picked blind out of three. The other one is whatever survived the host. So here is the sentence that matters. In two of the three cases the host had no choice at all. He was forced, and which door he was forced onto depended on where the car was. That is information, and switching is how you collect it. Which tells you exactly which assumption is doing the work, and gives us something to test. What if the host is not forced?"},{"title":"When the Host Does Not Know","startSeconds":276.5765625,"narration":"So change exactly one thing about the game. The host forgets where the car is. He still opens one of the two doors you did not pick, but now he chooses between them by flipping a coin. Which means that sometimes he opens the car himself. When that happens there is nothing left to decide, so we throw the round away and deal again. Everything else is untouched. Now count it again, the same way as before. Three places for the car, but each of those splits in two, because there are two doors his coin might send him to. Six branches. Same picture as before. Door one is yours, and the car is behind door one, door two or door three, exactly as it was. Start with the car behind door one. He flips, and opens door two: a goat, and staying wins. Or the coin lands the other way and he opens door three: also a goat, and staying wins again. Now the car is behind door two. If his coin sends him to door two, he has opened the car himself, and the round is void. If it sends him to door three, he shows a goat, and switching wins. And the car behind door three is the mirror of that. His coin sends him to door two, where he shows a goat, and switching wins. Or it sends him to door three, where he has shown the car, and the round is void again. Six branches, all equally likely, because the car is uniform and the coin is fair. Two of them end with the car standing there in the open, and those two get thrown out. So look at the four that survive. In two of them staying wins. In the other two switching wins. Two against two, and the advantage has gone. And look at what that does to your own door. In the first version it sat at one third whatever he did, because he could never fail to show a goat. Here, the fact that he happened to miss the car is itself evidence, and it lifts your door from one third to one half. Nothing else moved. Same doors, same first choice, same one in three that the car is behind door one at the start. The only thing that changed is whether the host knew where the car was. That is the real content of this puzzle, and it is why the first argument worked at all. A host who is forced two times in three is telling you something. A host who is flipping a coin is telling you nothing. So if the switching argument has always felt slippery, that is the joint it turns on. Now let me turn the numbers up until the thing is obvious."},{"title":"A Hundred Doors","startSeconds":429.0543541666667,"narration":"Last picture, and this is the one that usually finishes the argument. A hundred doors. One car, ninety nine goats. You pick the door in the corner there, and it stays picked. The chance that you found the car with that pick is one in a hundred. You will not argue with me about that, because there was nothing at all to go on when you chose. Now the host, who knows exactly where the car is, opens ninety eight of the other doors, and every single one of them has a goat behind it. He is not lucky. He is stepping around the car, ninety eight times in a row. Two doors are left. Yours, and the one door out of ninety nine that he walked past every single time. So ask the question in this form. Did your one in a hundred just turn into one in two? Of course it did not. Your door is still the door you picked blind out of a hundred. All the probability that was spread thinly over the other ninety nine has been swept, by a host who was never allowed to touch the car, onto the single door he left standing. And the three door game is that same game with the numbers turned down. Your door holds a third. The other two hold two thirds between them, and the host pours all of it onto whichever door he does not open. So here is the whole thing on one board. If the host knows where the car is and has to avoid it, switching wins two times in three. If he is opening doors at random and merely happened to miss the car, it is one half either way, and you may do whatever you like. Three things to keep. Your first door keeps whatever chance it had at the moment you picked it. A host who must avoid the car sweeps all the rest onto a single door. And a host who is guessing sweeps nothing at all. Which is why this puzzle was never really about the doors. It was about the host, and whether he was filtering. When he is, switch."}]}}