{"version":1,"lectureId":"01M1D5AZXKQQ0DPCPAC2B7B5JY","attempt":0,"publication":{"slug":"the-secretary-problem-picking-from-the-top-10-percent","title":"The Secretary Problem: Picking from the Top 10 Percent","subject":"mathematics","summary":"A hundred houses arrive one at a time in random order. You accept or reject each on the spot, and a rejected house is gone for good. Aiming for the single best house is famously hard, and even the optimal plan wins only about thirty seven times in a hundred. Aiming instead for any of the ten best changes everything. This lecture builds the cutoff rule by hand: look at thirty seven houses and buy nothing, take the best of them as your bar, then buy the first house that clears it. Counting on screen shows why that bar is almost always already in the top ten, why anything clearing it is then a winner with certainty, and how relaxing the bar near the end of the list removes the one way the plan can fail.","metaDescription":"A hundred houses, one at a time, no going back. Why looking at 37 and then taking the best you have seen lands you in the top ten.","transcript":"Somebody offers you a hundred houses. You will see them one at a time, in a random order, and each time you must say yes or no on the spot. Say no, and that house is gone for good. Here is the kind part. You do not need the best of the hundred. Any of the ten best would make you perfectly happy, and that single change turns a famously hard problem into an easy one. Let me build a scale to measure this on. Give every house a score, and hand the scores one to a hundred out to the hundred houses, one each. So a hundred is the best house there is, and one is the worst. They arrive in a random order, so they land anywhere along it. The first house scores forty two. The second scores sixty eight, better, and both are already gone, because you turned them down. One warning about this picture. We are outside the story, so we can read the scores. The buyer cannot. All the buyer can ever do is hold two houses side by side and say which of them is better. Now the target. The ten best houses are the ten highest scores, ninety one up to a hundred, so winning means landing under this brace at the top of the scale. Compare that with the famous version, where nothing but the very best house will do. That target is a single point, and everything else is failure. The known answer is to look at thirty seven houses, buy nothing, then take the first one that beats them all. It wins about thirty seven times in a hundred. Now widen the target to ten points and keep that very same strategy, cutoff and all. Whenever it buys a house, the house it buys is one of the ten best better than ninety eight times in a hundred. Making sure it buys at all is the only job left over. Here is one journey through a hundred houses, seen from outside. Along the bottom, the order they arrive in. Up the side, their scores, with the ten best of them inside the red band at the top. I will not draw all hundred houses. I will draw one thing instead: the best score seen so far. That line can only climb, and every step in it is a house that beat everything before it. There are only five or six of those steps in a random hundred, and they thin out as you go, because beating the best of fifty is far harder than beating the best of five. Now the strategy, in three lines. One: let the first thirty seven houses go past, whatever they are, and buy nothing at all. Two: take the best score among those thirty seven. Here it is, ninety three, at house twenty six. That score is now your bar. Three: from house thirty eight onward, buy the first house that clears the bar. And that is the whole rule. Look for a third of the list, then take the first house that beats everything you saw while you were looking. Watch it run. House forty one is pleasant enough. House forty nine scores eighty seven, better than almost anything you have seen, and you must let it go, because it does not clear the bar. Then house fifty eight scores ninety seven. That clears the bar, so you buy it and stop. And look where it landed, inside the band. It is the fourth best house of the hundred. The very best house of all, the one scoring a hundred, turns up much later, at house eighty four. You never see it, and it does not matter in the slightest. You were not trying to find it. Two things could have gone wrong, and both are about where that dashed line stands. Stop looking after five houses and the bar would sit at sixty eight, low enough for an ordinary house to clear it. Look at ninety and the bar would be a hundred, which nothing left could ever beat. So why is a bar built from only thirty seven houses good enough? Let me take the climbing line away and mark the ten best houses instead. There they are. Ten houses inside the band, landing anywhere, because the order is random. Now count how many arrived in your first thirty seven. Exactly one, the ninety three that set your bar. One is all you need, and that is the secret. Each of these ten, on its own, has a sixty three percent chance of landing after the line. For your bar to miss the band, every single one of them has to. Sixty three percent, ten times over, is about one chance in a hundred. The ten crowd each other slightly, so the true figure is smaller still, about one chance in a hundred and thirty five. Which means the bar you built out of thirty seven houses is itself one of the ten best, ninety nine times in a hundred. And here is the step that turns that into a promise. Suppose the bar is inside the band. The band runs from the bar up to the top of the scale, so every score above the bar is inside the band as well. So any house clearing the bar is one of the ten best. Not probably. Certainly. The only way this rule hands you a bad house is if the bar itself missed the band, and that is the one chance in a hundred and thirty five. Which leaves exactly one weakness, and it is not about buying badly. It is about never buying at all. Here is a different hundred houses, in a different order, and this time the luck runs the other way. The best house in the whole set, the one scoring a hundred, arrives at house twelve, while you are still only looking. So the best of your first thirty seven is a hundred, and your bar sits at the very top of the scale. Nothing can clear it. You watch the remaining sixty three houses go past, turn every one of them down, and reach the end of the list having bought nothing at all. And this is no freak. The best house of the hundred is equally likely to be anywhere, so it lands inside your looking window thirty seven times in a hundred, and every one of those runs ends empty handed. Here is the cure, drawn as a second picture. Along the bottom, the order again. Up the side, how many houses you are willing to have above the one you buy. At first you accept nothing but the best you have seen. As the list runs out you relax. After fifty, the second best of what you have seen will do, then the third, and near the end you take almost anything decent. Why is that safe? The second best of fifty houses misses the top ten only if nine or ten of the ten best hid in the second half. That is like ten coin flips coming up almost all heads, about eleven chances in a thousand. Relax that way all the way down the list, and the rule nearly always buys, and what it buys is nearly always in the top ten. Better than ninety eight times in a hundred, and that is the promise. So here is the whole strategy, in four lines. Look at the first thirty seven houses and buy nothing. Set the bar at the best of them. Then buy the first house that clears it. And as the list runs out, let the bar come down, one place at a time, so you never reach the end with nothing. That is the whole of it. You cannot know which house is the best. But you can spend a third of your search learning what good looks like, and then have the nerve to stop.","watch":{"version":1,"scenes":[{"title":"The Offer, and the Target","start":0,"end":110.89575000000002,"objects":{"classic":"a Math [text] that says \"$P(upright(\"the single best\")) approx 0.37$\"","house_one":"a Point [blue] labelled \"42\" drawn in scale (location=(42.0, 0.0))","house_two":"a Point [blue] labelled \"68\" drawn in scale (location=(68.0, 0.0))","ours":"a Math [text] that says \"$P(upright(\"top ten\")) > 0.98$\"","point":"a Point [yellow] drawn in scale (location=(83.0, 0.0))","point_2":"a Point [yellow] drawn in scale (location=(55.0, 0.0))","rules":"a Panel that says \"A hundred houses arrive one at a time, in random order. Accept or reject each one on the spot, and a rejected house is gone for good. You do not need the best house. You need one of the ten best.\"","scale":"a NumberLine labelled \"upright(\"score\")\" (x_range=(0.0, 104.0), include_numbers=True, ticks_every=20.0)","single_best":"a Point [green] labelled \"upright(\"the best\")\" drawn in scale (location=(100.0, 0.0))","ten_best":"a Brace [text] labelled \"upright(\"the ten best\")\" drawn in scale (x_start=90.5, x_end=100.5)"},"beats":[{"start":0,"say":"Somebody offers you a hundred houses. You will see them one at a time, in a random order, and each time you must say yes or no on the spot. Say no, and that house is gone for good.","live":[],"does":[[0,"rules is shown on the screen, written out."]]},{"start":12.5,"say":"Here is the kind part. You do not need the best of the hundred. Any of the ten best would make you perfectly happy, and that single change turns a famously hard problem into an easy one.","live":["rules"],"does":[[17.933,"rules (the \"one of the ten best\" part) is emphasized."],[21.149,"rules (the \"one of the ten best\" part) is no longer emphasized."],[24.4235,"rules moves to a new place on the board."]]},{"start":25.0235,"say":"Let me build a scale to measure this on. Give every house a score, and hand the scores one to a hundred out to the hundred houses, one each. So a hundred is the best house there is, and one is the worst.","live":null,"does":[[25.0235,"scale is shown on the screen, written out."]]},{"start":38.707499999999996,"say":"They arrive in a random order, so they land anywhere along it. The first house scores forty two. The second scores sixty eight, better, and both are already gone, because you turned them down.","live":["rules","scale"],"does":[[44.047999999999995,"house_one is shown on the screen, written out."],[46.404999999999994,"house_two is shown on the screen, written out."]]},{"start":51.7885,"say":"One warning about this picture. We are outside the story, so we can read the scores. The buyer cannot. All the buyer can ever do is hold two houses side by side and say which of them is better.","live":["rules","scale","house_one","house_two"],"does":[[61.56399999999999,"point is shown on the screen, grown."],[63.69376983279971,"point is hidden from the screen."],[63.839999999999996,"point_2 is shown on the screen, grown."]]},{"start":65.218,"say":"Now the target. The ten best houses are the ten highest scores, ninety one up to a hundred, so winning means landing under this brace at the top of the scale.","live":["rules","scale","house_one","house_two","point_2"],"does":[[65.218,"house_one is hidden from the screen."],[65.218,"house_two is hidden from the screen."],[66.6241280671054,"point_2 is hidden from the screen."],[69.676,"The segment (90.5, 0.0) to (100.5, 0.0) in scale is lit up."],[72.96199999999999,"ten_best is shown on the screen, written out."],[72.96199999999999,"scale: retire a lit segment (unemphasize_line)."]]},{"start":75.33800000000001,"say":"Compare that with the famous version, where nothing but the very best house will do. That target is a single point, and everything else is failure. The known answer is to look at thirty seven houses, buy nothing, then take the first one that beats them all. It wins about thirty seven times in a hundred.","live":["rules","scale","ten_best"],"does":[[82.14200000000001,"single_best is shown on the screen, written out."],[92.254,"scale moves to a new place on the board."],[92.254,"classic is shown on the screen, written out."]]},{"start":94.8045,"say":"Now widen the target to ten points and keep that very same strategy, cutoff and all. Whenever it buys a house, the house it buys is one of the ten best better than ninety eight times in a hundred. Making sure it buys at all is the only job left over.","live":["rules","scale","classic","ten_best","single_best"],"does":[[100.308,"ours is shown on the screen, written out."],[104.267,"A box is drawn around ours."],[109.85408333333334,"classic is hidden from the screen — left the board."],[109.85408333333334,"ours is hidden from the screen — left the board."],[109.85408333333334,"rules is hidden from the screen — left the board."],[109.85408333333334,"scale is hidden from the screen — left the board."],[109.85408333333334,"ten_best is hidden from the screen — scale left the board."],[109.85408333333334,"single_best is hidden from the screen — scale left the board."]]}]},{"title":"Look, Then Leap","start":110.89575000000002,"end":328.6203958333333,"objects":{"axes":"an Axes (x_range=(0.0, 104.0), y_range=(0.0, 108.0), x_ticks_every=20.0)","band":"a Polygon [red] drawn in axes (vertices=((0.0, 90.5), (104.0, 90.5), (104.0, 100.5), (0.0, 100.5)), fill_opacity=0.25)","bar":"a Line [yellow] labelled \"upright(\"bar\")\" drawn in axes (start=(37.0, 93.0), end=(104.0, 93.0))","bought":"a Point [green] labelled \"97\" drawn in axes (location=(58.0, 97.0))","cut":"a Line [gray] drawn in axes (start=(37.0, 0.0), end=(37.0, 104.0), dashed=True)","exact":"a Math [text] that says \"$frac(63 dot.op dots.h dot.op 54, 100 dot.op dots.h dot.op 91) = 0.0074$\"","good":"a Math [text] that says \"$P(upright(\"bar in the band\")) = 0.9926$\"","head_rule":"a Heading that says \"Look, Then Leap\"","head_why":"a Heading that says \"Why That Bar Is High Enough\"","outcome":"a Math [text] that says \"$upright(\"bought\") = 97$\"","overall_best":"a Point [gray] labelled \"100\" drawn in axes (location=(84.0, 100.0))","point":"a Point [yellow] drawn in axes (location=(11.0, 88.0))","point_2":"a Point [yellow] drawn in axes (location=(41.0, 62.0))","point_3":"a Point [yellow] drawn in axes (location=(49.0, 87.0))","rough":"a Math [text] that says \"$(63 / 100)^10 approx 0.010$\"","running":"a FunctionPlot [blue] labelled \"upright(\"best so far\")\" drawn in axes (function=<function>, x_range=(1.0, 100.0))","safe":"a Polygon [green] drawn in axes (vertices=((37.0, 93.0), (104.0, 93.0), (104.0, 100.5), (37.0, 100.5)), fill_opacity=0.4)","sample_best":"a Point [yellow] labelled \"93\" drawn in axes (location=(26.0, 93.0))","step_one":"a Text [text] that says \"1. Look at the first 37. Buy nothing.\"","step_three":"a Text [text] that says \"3. Buy the first house that clears it.\"","step_two":"a Text [text] that says \"2. The bar: the best of those 37.\"","top_dots":"a Point [red] drawn in axes (location=(26.0, 93.0))","top_dots_10":"a Point [red] drawn in axes (location=(95.0, 99.0))","top_dots_2":"a Point [red] drawn in axes (location=(39.0, 91.0))","top_dots_3":"a Point [red] drawn in axes (location=(44.0, 92.0))","top_dots_4":"a Point [red] drawn in axes (location=(58.0, 97.0))","top_dots_5":"a Point [red] drawn in axes (location=(63.0, 94.0))","top_dots_6":"a Point [red] drawn in axes (location=(71.0, 95.0))","top_dots_7":"a Point [red] drawn in axes (location=(79.0, 96.0))","top_dots_8":"a Point [red] drawn in axes (location=(84.0, 100.0))","top_dots_9":"a Point [red] drawn in axes (location=(88.0, 98.0))"},"beats":[{"start":110.89575000000002,"say":"Here is one journey through a hundred houses, seen from outside. Along the bottom, the order they arrive in. Up the side, their scores, with the ten best of them inside the red band at the top.","live":[],"does":[[110.89575000000002,"head_rule is shown on the screen, written out."],[110.89575000000002,"axes is shown on the screen, written out."],[122.71475000000002,"band is shown on the screen, written out."]]},{"start":124.97475000000001,"say":"I will not draw all hundred houses. I will draw one thing instead: the best score seen so far. That line can only climb, and every step in it is a house that beat everything before it.","live":["axes","head_rule","band"],"does":[[130.81475,"running is shown on the screen, written out."]]},{"start":139.16975000000002,"say":"There are only five or six of those steps in a random hundred, and they thin out as you go, because beating the best of fifty is far harder than beating the best of five.","live":["axes","head_rule","band","running"],"does":[[141.14375,"point is shown on the screen, grown."],[143.03934414732595,"point is hidden from the screen."]]},{"start":148.93025000000003,"say":"Now the strategy, in three lines. One: let the first thirty seven houses go past, whatever they are, and buy nothing at all.","live":null,"does":[[152.52975000000004,"axes moves to a new place on the board."],[152.52975000000004,"step_one is shown on the screen, written out."],[154.02775000000003,"cut is shown on the screen, written out."]]},{"start":159.07375000000002,"say":"Two: take the best score among those thirty seven. Here it is, ninety three, at house twenty six. That score is now your bar.","live":["step_one","axes","head_rule","band","running","cut"],"does":[[159.47975000000002,"step_two is shown on the screen, written out."],[163.97275000000002,"sample_best is shown on the screen, written out."],[168.25775000000002,"bar is shown on the screen, written out."]]},{"start":169.83275000000003,"say":"Three: from house thirty eight onward, buy the first house that clears the bar. And that is the whole rule. Look for a third of the list, then take the first house that beats everything you saw while you were looking.","live":["step_one","step_two","axes","head_rule","band","running","cut","sample_best","bar"],"does":[[170.30875000000003,"step_three is shown on the screen, written out."]]},{"start":183.80725,"say":"Watch it run. House forty one is pleasant enough. House forty nine scores eighty seven, better than almost anything you have seen, and you must let it go, because it does not clear the bar.","live":["step_one","step_two","step_three","axes","head_rule","band","running","cut","sample_best","bar"],"does":[[185.79275,"point_2 is shown on the screen, grown."],[188.0858917847197,"point_2 is hidden from the screen."],[188.16075,"point_3 is shown on the screen, grown."],[190.50510592340825,"point_3 is hidden from the screen."]]},{"start":195.94725,"say":"Then house fifty eight scores ninety seven. That clears the bar, so you buy it and stop. And look where it landed, inside the band. It is the fourth best house of the hundred.","live":null,"does":[[197.80475,"bought is shown on the screen, written out."],[200.59175,"outcome is shown on the screen, written out."],[202.71575,"bought is indicated — a transient flash."],[205.36275,"A box is drawn around outcome."]]},{"start":207.73925000000003,"say":"The very best house of all, the one scoring a hundred, turns up much later, at house eighty four. You never see it, and it does not matter in the slightest. You were not trying to find it.","live":["step_one","step_two","step_three","outcome","axes","head_rule","band","running","cut","sample_best","bar","bought"],"does":[[212.39475000000002,"overall_best is shown on the screen, written out."]]},{"start":219.31125000000003,"say":"Two things could have gone wrong, and both are about where that dashed line stands. Stop looking after five houses and the bar would sit at sixty eight, low enough for an ordinary house to clear it. 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Ten houses inside the band, landing anywhere, because the order is random. Now count how many arrived in your first thirty seven. Exactly one, the ninety three that set your bar.","live":["axes","band","cut","bar","head_why","top_dots","top_dots_2","top_dots_3","top_dots_4","top_dots_5","top_dots_6","top_dots_7","top_dots_8","top_dots_9","top_dots_10"],"does":[[254.86075000000005,"top_dots is indicated — a transient flash."]]},{"start":259.08325,"say":"One is all you need, and that is the secret. Each of these ten, on its own, has a sixty three percent chance of landing after the line. For your bar to miss the band, every single one of them has to.","live":null,"does":[[270.8207500000001,"rough is shown on the screen, written out."]]},{"start":273.45275000000004,"say":"Sixty three percent, ten times over, is about one chance in a hundred. The ten crowd each other slightly, so the true figure is smaller still, about one chance in a hundred and thirty five.","live":["axes","band","cut","bar","rough","head_why","top_dots","top_dots_2","top_dots_3","top_dots_4","top_dots_5","top_dots_6","top_dots_7","top_dots_8","top_dots_9","top_dots_10"],"does":[[281.11575000000005,"exact is shown on the screen, written out."]]},{"start":285.98825,"say":"Which means the bar you built out of thirty seven houses is itself one of the ten best, ninety nine times in a hundred.","live":["axes","band","cut","bar","rough","exact","head_why","top_dots","top_dots_2","top_dots_3","top_dots_4","top_dots_5","top_dots_6","top_dots_7","top_dots_8","top_dots_9","top_dots_10"],"does":[[291.32875,"good is shown on the screen, written out."]]},{"start":293.61225,"say":"And here is the step that turns that into a promise. Suppose the bar is inside the band. 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Near the end, let the bar come down.\"","step_one":"a Text [text] that says \"1. Look at the first 37 houses. Buy nothing.\"","step_three":"a Text [text] that says \"3. Buy the first house that clears the bar.\"","step_two":"a Text [text] that says \"2. Set the bar at the best of those 37.\"","unlucky":"a FunctionPlot [blue] labelled \"upright(\"best so far\")\" drawn in axes (function=<function>, x_range=(1.0, 100.0))"},"beats":[{"start":328.6203958333333,"say":"Here is a different hundred houses, in a different order, and this time the luck runs the other way.","live":[],"does":[[328.6203958333333,"head_hole is shown on the screen, written out."],[328.6203958333333,"axes is shown on the screen, written out."],[330.76839583333333,"band is shown on the screen, written out."],[332.9513958333333,"unlucky is shown on the screen, written out."]]},{"start":335.1298958333333,"say":"The best house in the whole set, the one scoring a hundred, arrives at house twelve, while you are still only looking. So the best of your first thirty seven is a hundred, and your bar sits at the very top of the scale.","live":["axes","head_hole","band","unlucky"],"does":[[339.5413958333333,"peak is shown on the screen, written out."],[344.1853958333333,"cut is shown on the screen, written out."],[346.1013958333333,"The segment (37.0, 100.0) to (104.0, 100.0) in axes is lit up."],[348.5043958333333,"axes: retire a lit segment (unemphasize_line)."]]},{"start":349.1043958333333,"say":"Nothing can clear it. You watch the remaining sixty three houses go past, turn every one of them down, and reach the end of the list having bought nothing at all.","live":["axes","head_hole","band","unlucky","peak","cut"],"does":[[353.1093958333333,"point is shown on the screen, grown."],[354.96161363275456,"point is hidden from the screen."]]},{"start":358.5508958333333,"say":"And this is no freak. The best house of the hundred is equally likely to be anywhere, so it lands inside your looking window thirty seven times in a hundred, and every one of those runs ends empty handed.","live":null,"does":[[365.21539583333333,"axes moves to a new place on the board."],[365.21539583333333,"hole is shown on the screen, written out."],[369.5693958333333,"axes is hidden from the screen — left the board."],[369.5693958333333,"band is hidden from the screen — axes left the board."],[369.5693958333333,"unlucky is hidden from the screen — axes left the board."],[369.5693958333333,"peak is hidden from the screen — axes left the board."],[369.5693958333333,"cut is hidden from the screen — axes left the board."],[369.5693958333333,"head_hole is hidden from the screen — left the board."],[369.5693958333333,"hole is hidden from the screen — left the board."]]},{"start":370.1693958333333,"say":"Here is the cure, drawn as a second picture. Along the bottom, the order again. Up the side, how many houses you are willing to have above the one you buy.","live":[],"does":[[370.1693958333333,"head_fix is shown on the screen, written out."],[372.4913958333333,"ladder is shown on the screen, written out."],[378.4243958333333,"relax is shown on the screen, written out."]]},{"start":381.1598958333333,"say":"At first you accept nothing but the best you have seen. As the list runs out you relax. After fifty, the second best of what you have seen will do, then the third, and near the end you take almost anything decent.","live":["ladder","head_fix","relax"],"does":[[386.2453958333333,"relax is redrawn as the numbers it depends on change."],[386.2453958333333,"drawn ticks to 100.0."]]},{"start":395.2163958333333,"say":"Why is that safe? The second best of fifty houses misses the top ten only if nine or ten of the ten best hid in the second half. That is like ten coin flips coming up almost all heads, about eleven chances in a thousand.","live":null,"does":[[407.5573958333333,"ladder moves to a new place on the board."],[407.5573958333333,"coins is shown on the screen, written out."]]},{"start":410.0843958333333,"say":"Relax that way all the way down the list, and the rule nearly always buys, and what it buys is nearly always in the top ten. Better than ninety eight times in a hundred, and that is the promise.","live":["coins","ladder","head_fix","relax"],"does":[[414.09039583333333,"promise is shown on the screen, written out."],[421.1428958333333,"coins is hidden from the screen — left the board."],[421.1428958333333,"head_fix is hidden from the screen — left the board."],[421.1428958333333,"ladder is hidden from the screen — left the board."],[421.1428958333333,"relax is hidden from the screen — ladder left the board."],[421.1428958333333,"promise is hidden from the screen — left the board."]]},{"start":421.7428958333333,"say":"So here is the whole strategy, in four lines. Look at the first thirty seven houses and buy nothing. Set the bar at the best of them. Then buy the first house that clears it.","live":[],"does":[[421.7428958333333,"head_recap is shown on the screen, written out."],[424.7093958333333,"step_one is shown on the screen, written out."],[427.5423958333333,"step_two is shown on the screen, written out."],[430.6533958333333,"step_three is shown on the screen, written out."]]},{"start":432.5308958333333,"say":"And as the list runs out, let the bar come down, one place at a time, so you never reach the end with nothing.","live":["step_one","step_two","step_three","head_recap"],"does":[[435.11939583333333,"step_four is shown on the screen, written out."]]},{"start":439.7718958333333,"say":"That is the whole of it. You cannot know which house is the best. But you can spend a third of your search learning what good looks like, and then have the nerve to stop.","live":["step_one","step_two","step_three","step_four","head_recap"],"does":[[448.73812499999997,"head_recap is hidden from the screen — left the board."],[448.73812499999997,"step_four is hidden from the screen — left the board."],[448.73812499999997,"step_one is hidden from the screen — left the board."],[448.73812499999997,"step_three is hidden from the screen — left the board."],[448.73812499999997,"step_two is hidden from the screen — left the board."]]}]}]},"durationSeconds":450,"chapters":[{"title":"The Offer, and the Target","startSeconds":0,"narration":"Somebody offers you a hundred houses. You will see them one at a time, in a random order, and each time you must say yes or no on the spot. Say no, and that house is gone for good. Here is the kind part. You do not need the best of the hundred. Any of the ten best would make you perfectly happy, and that single change turns a famously hard problem into an easy one. Let me build a scale to measure this on. Give every house a score, and hand the scores one to a hundred out to the hundred houses, one each. So a hundred is the best house there is, and one is the worst. They arrive in a random order, so they land anywhere along it. The first house scores forty two. The second scores sixty eight, better, and both are already gone, because you turned them down. One warning about this picture. We are outside the story, so we can read the scores. The buyer cannot. All the buyer can ever do is hold two houses side by side and say which of them is better. Now the target. The ten best houses are the ten highest scores, ninety one up to a hundred, so winning means landing under this brace at the top of the scale. Compare that with the famous version, where nothing but the very best house will do. That target is a single point, and everything else is failure. The known answer is to look at thirty seven houses, buy nothing, then take the first one that beats them all. It wins about thirty seven times in a hundred. Now widen the target to ten points and keep that very same strategy, cutoff and all. Whenever it buys a house, the house it buys is one of the ten best better than ninety eight times in a hundred. Making sure it buys at all is the only job left over."},{"title":"Look, Then Leap","startSeconds":110.89575000000002,"narration":"Here is one journey through a hundred houses, seen from outside. Along the bottom, the order they arrive in. Up the side, their scores, with the ten best of them inside the red band at the top. I will not draw all hundred houses. I will draw one thing instead: the best score seen so far. That line can only climb, and every step in it is a house that beat everything before it. There are only five or six of those steps in a random hundred, and they thin out as you go, because beating the best of fifty is far harder than beating the best of five. Now the strategy, in three lines. One: let the first thirty seven houses go past, whatever they are, and buy nothing at all. Two: take the best score among those thirty seven. Here it is, ninety three, at house twenty six. That score is now your bar. Three: from house thirty eight onward, buy the first house that clears the bar. And that is the whole rule. Look for a third of the list, then take the first house that beats everything you saw while you were looking. Watch it run. House forty one is pleasant enough. House forty nine scores eighty seven, better than almost anything you have seen, and you must let it go, because it does not clear the bar. Then house fifty eight scores ninety seven. That clears the bar, so you buy it and stop. And look where it landed, inside the band. It is the fourth best house of the hundred. The very best house of all, the one scoring a hundred, turns up much later, at house eighty four. You never see it, and it does not matter in the slightest. You were not trying to find it. Two things could have gone wrong, and both are about where that dashed line stands. Stop looking after five houses and the bar would sit at sixty eight, low enough for an ordinary house to clear it. Look at ninety and the bar would be a hundred, which nothing left could ever beat. So why is a bar built from only thirty seven houses good enough? Let me take the climbing line away and mark the ten best houses instead. There they are. Ten houses inside the band, landing anywhere, because the order is random. Now count how many arrived in your first thirty seven. Exactly one, the ninety three that set your bar. One is all you need, and that is the secret. Each of these ten, on its own, has a sixty three percent chance of landing after the line. For your bar to miss the band, every single one of them has to. Sixty three percent, ten times over, is about one chance in a hundred. The ten crowd each other slightly, so the true figure is smaller still, about one chance in a hundred and thirty five. Which means the bar you built out of thirty seven houses is itself one of the ten best, ninety nine times in a hundred. And here is the step that turns that into a promise. Suppose the bar is inside the band. The band runs from the bar up to the top of the scale, so every score above the bar is inside the band as well. So any house clearing the bar is one of the ten best. Not probably. Certainly. The only way this rule hands you a bad house is if the bar itself missed the band, and that is the one chance in a hundred and thirty five. Which leaves exactly one weakness, and it is not about buying badly. It is about never buying at all."},{"title":"The Hole, and How to Close It","startSeconds":328.6203958333333,"narration":"Here is a different hundred houses, in a different order, and this time the luck runs the other way. The best house in the whole set, the one scoring a hundred, arrives at house twelve, while you are still only looking. So the best of your first thirty seven is a hundred, and your bar sits at the very top of the scale. Nothing can clear it. You watch the remaining sixty three houses go past, turn every one of them down, and reach the end of the list having bought nothing at all. And this is no freak. The best house of the hundred is equally likely to be anywhere, so it lands inside your looking window thirty seven times in a hundred, and every one of those runs ends empty handed. Here is the cure, drawn as a second picture. Along the bottom, the order again. Up the side, how many houses you are willing to have above the one you buy. At first you accept nothing but the best you have seen. As the list runs out you relax. After fifty, the second best of what you have seen will do, then the third, and near the end you take almost anything decent. Why is that safe? The second best of fifty houses misses the top ten only if nine or ten of the ten best hid in the second half. That is like ten coin flips coming up almost all heads, about eleven chances in a thousand. Relax that way all the way down the list, and the rule nearly always buys, and what it buys is nearly always in the top ten. Better than ninety eight times in a hundred, and that is the promise. So here is the whole strategy, in four lines. Look at the first thirty seven houses and buy nothing. Set the bar at the best of them. Then buy the first house that clears it. And as the list runs out, let the bar come down, one place at a time, so you never reach the end with nothing. That is the whole of it. You cannot know which house is the best. But you can spend a third of your search learning what good looks like, and then have the nerve to stop."}]}}