WEBVTT 00:00:07.751 --> 00:00:09.491 Maybe the fates got clumsy. 00:00:09.491 --> 00:00:12.311 Maybe Poseidon had one of his angry days. 00:00:12.311 --> 00:00:15.621 However it happened, the underworld is overcrowded, 00:00:15.621 --> 00:00:20.157 and Zeus has ordered Hades to let some spirits out. 00:00:20.157 --> 00:00:24.599 Hades arranges all the souls of the dead in a line before Cerberus. 00:00:24.599 --> 00:00:28.489 When one of his three heads bites down on the soul in front of it, 00:00:28.489 --> 00:00:31.219 they’ll get returned to the land of the living. 00:00:31.219 --> 00:00:36.196 Anyone to the left must get out of line and stay in Hades forever. 00:00:36.196 --> 00:00:41.698 And everyone else shuffles forward, at which point Cerberus will feed again. 00:00:41.698 --> 00:00:47.280 Each of the dog’s heads has an equal chance of being the one to bite each time, 00:00:47.280 --> 00:00:51.280 and no two ever bite simultaneously. 00:00:51.280 --> 00:00:55.398 Unfortunately, Hades’ minions forgot to tell you what was happening, 00:00:55.398 --> 00:01:00.141 and by the time you show up there are only 99 souls left in line. 00:01:00.141 --> 00:01:04.757 Hades looks furious and drawing attention to yourself won’t end well. 00:01:04.757 --> 00:01:09.613 But suddenly, time freezes, and Hermes steps out of the shadows. 00:01:09.613 --> 00:01:12.953 He tells you he can instantly put you into the line, 00:01:12.953 --> 00:01:15.513 and no one will realize what happened. 00:01:15.513 --> 00:01:19.513 But he’ll only grant his grace to someone clever enough 00:01:19.513 --> 00:01:21.373 to take full advantage of it. 00:01:21.373 --> 00:01:24.483 Choose the best place in line and he’ll give you the spot. 00:01:24.483 --> 00:01:27.993 Choose wrong, and he’ll leave you to rot. 00:01:27.993 --> 00:01:29.823 Which spot should you pick? 00:01:29.823 --> 00:01:31.833 Pause the video to figure it out yourself. 00:01:31.833 --> 00:01:32.533 Answer in 3 00:01:32.533 --> 00:01:35.063 Answer in 2 00:01:35.063 --> 00:01:37.673 Answer in 1 00:01:37.673 --> 00:01:41.745 It’s possible to calculate the exact probability of going free 00:01:41.745 --> 00:01:43.875 in all 100 spots. 00:01:43.875 --> 00:01:46.605 But there’s a much simpler path to the solution 00:01:46.605 --> 00:01:49.605 that requires surprisingly little calculation. 00:01:49.605 --> 00:01:52.315 Imagine being anywhere in line. 00:01:52.315 --> 00:01:56.406 Way up at the front, one of the three heads will pick someone at random, 00:01:56.406 --> 00:01:59.896 and you’ll move forward 1, 2, or 3 spaces. 00:01:59.896 --> 00:02:02.226 Since each is equally likely, 00:02:02.226 --> 00:02:05.486 your chance of survival from wherever you started 00:02:05.486 --> 00:02:10.525 is the average of the chances from each of the three spaces in front of you. 00:02:10.525 --> 00:02:14.751 And this is where you can find a huge shortcut. 00:02:14.751 --> 00:02:20.804 Averages must be on or between the extremes of what you’re averaging–– 00:02:20.804 --> 00:02:25.500 they can never be higher than the highest value or lower than the lowest. 00:02:25.500 --> 00:02:29.351 So whatever your chances of survival are where you start, 00:02:29.351 --> 00:02:33.846 one of the three places in front of you is at least as good, 00:02:33.846 --> 00:02:34.986 and probably better. 00:02:34.986 --> 00:02:38.246 This observation is incredibly powerful. 00:02:38.246 --> 00:02:41.126 It means that wherever you are in line, 00:02:41.126 --> 00:02:45.993 it’d be wise to trade your place for one of the three spots in front of you. 00:02:45.993 --> 00:02:50.177 Let’s ignore which for now and think of them as a trio–– 00:02:50.177 --> 00:02:54.958 this trio’s maximum value is better than this trio’s, and so on. 00:02:54.958 --> 00:02:57.356 Keep going and you’ll reach the front... 00:02:57.356 --> 00:03:01.938 These three spots must contain the extreme values–– 00:03:01.938 --> 00:03:04.218 the best and worst probabilities–– 00:03:04.218 --> 00:03:05.466 for the entire line. NOTE Paragraph 00:03:05.466 --> 00:03:09.928 In other words, they’re all we need to consider. 00:03:09.928 --> 00:03:11.248 Place 1 is bad. 00:03:11.248 --> 00:03:14.718 Head one would save you, and the other two doom you forever. 00:03:14.718 --> 00:03:17.508 That’s just a 1 in 3 chance to escape. 00:03:17.508 --> 00:03:21.058 Place 2 is better: head two is great, 00:03:21.058 --> 00:03:26.572 head 3 is bad, and head 1 is ok in that it gives you another chance. 00:03:26.572 --> 00:03:31.324 But place 3 is best, because head 3 saves you 00:03:31.324 --> 00:03:35.681 while heads 1 and 2 both give you extra chances. 00:03:35.681 --> 00:03:39.331 If you did want to consider the exact probabilities, 00:03:39.331 --> 00:03:45.568 the odds of surviving in place 3 are 16 out of 27, or close to 60%. 00:03:45.568 --> 00:03:51.616 The spots later in line tend to be very close to having a 50% chance of survival. 00:03:51.616 --> 00:03:52.936 Why 50%? 00:03:52.936 --> 00:03:57.204 Because every time Cerberus sends one soul up to be reborn, 00:03:57.204 --> 00:04:01.571 he leaves 0, 1, or 2 souls in the underworld. 00:04:01.571 --> 00:04:07.040 That averages out to one person staying for each one who gets freed. 00:04:07.040 --> 00:04:11.105 But you can beat those odds handily with what you now know. 00:04:11.105 --> 00:04:14.055 Hermes has places to be, and so do you. 00:04:14.055 --> 00:04:18.055 He rewards your insight by sneaking you into the third spot. 00:04:18.055 --> 00:04:22.405 And from there it’ll be just a short wait to learn your ultimate fate.