1
00:00:07,751 --> 00:00:09,491
Maybe the fates got clumsy.

2
00:00:09,491 --> 00:00:12,311
Maybe Poseidon had one of his angry days.

3
00:00:12,311 --> 00:00:15,621
However it happened, 
the underworld is overcrowded,

4
00:00:15,621 --> 00:00:20,157
and Zeus has ordered Hades
to let some spirits out.

5
00:00:20,157 --> 00:00:24,599
Hades arranges all the souls
of the dead in a line before Cerberus.

6
00:00:24,599 --> 00:00:28,489
When one of his three heads bites 
down on the soul in front of it,

7
00:00:28,489 --> 00:00:31,219
they’ll get returned 
to the land of the living.

8
00:00:31,219 --> 00:00:36,196
Anyone to the left must get out 
of line and stay in Hades forever.

9
00:00:36,196 --> 00:00:41,698
And everyone else shuffles forward,
at which point Cerberus will feed again.

10
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,

11
00:00:47,280 --> 00:00:51,280
and no two ever bite simultaneously.

12
00:00:51,280 --> 00:00:55,398
Unfortunately, Hades’ minions forgot
to tell you what was happening,

13
00:00:55,398 --> 00:01:00,141
and by the time you show up 
there are only 99 souls left in line.

14
00:01:00,141 --> 00:01:04,757
Hades looks furious and drawing attention
to yourself won’t end well.

15
00:01:04,757 --> 00:01:09,613
But suddenly, time freezes, 
and Hermes steps out of the shadows.

16
00:01:09,613 --> 00:01:12,953
He tells you he can instantly put 
you into the line,

17
00:01:12,953 --> 00:01:15,513
and no one will realize what happened.

18
00:01:15,513 --> 00:01:19,513
But he’ll only grant his grace 
to someone clever enough

19
00:01:19,513 --> 00:01:21,373
to take full advantage of it.

20
00:01:21,373 --> 00:01:24,483
Choose the best place in line 
and he’ll give you the spot.

21
00:01:24,483 --> 00:01:27,993
Choose wrong, and he’ll leave you to rot.

22
00:01:27,993 --> 00:01:29,823
Which spot should you pick?

23
00:01:29,823 --> 00:01:31,833
Pause the video to figure it out yourself.

24
00:01:31,833 --> 00:01:32,533
Answer in 3

25
00:01:32,533 --> 00:01:35,063
Answer in 2

26
00:01:35,063 --> 00:01:37,673
Answer in 1

27
00:01:37,673 --> 00:01:41,745
It’s possible to calculate the exact 
probability of going free

28
00:01:41,745 --> 00:01:43,875
in all 100 spots.

29
00:01:43,875 --> 00:01:46,605
But there’s a much simpler path 
to the solution

30
00:01:46,605 --> 00:01:49,765
that requires surprisingly
little calculation.

31
00:01:49,765 --> 00:01:52,315
Imagine being anywhere in line.

32
00:01:52,315 --> 00:01:56,406
Way up at the front, one of the three
heads will pick someone at random,

33
00:01:56,406 --> 00:01:59,896
and you’ll move forward 1, 2, or 3 spaces.

34
00:01:59,896 --> 00:02:02,226
Since each is equally likely,

35
00:02:02,226 --> 00:02:05,486
your chance of survival 
from wherever you started

36
00:02:05,486 --> 00:02:10,745
is the average of the chances from
each of the three spaces in front of you.

37
00:02:10,745 --> 00:02:15,191
And this is where you can
find a huge shortcut.

38
00:02:15,191 --> 00:02:20,804
Averages must be on or between 
the extremes of what you’re averaging—

39
00:02:20,804 --> 00:02:23,730
they can never be higher 
than the highest value

40
00:02:23,730 --> 00:02:25,780
or lower than the lowest.

41
00:02:25,780 --> 00:02:29,351
So whatever your chances of survival
are where you start,

42
00:02:29,351 --> 00:02:33,846
one of the three places in front 
of you is at least as good,

43
00:02:33,846 --> 00:02:35,226
and probably better.

44
00:02:35,226 --> 00:02:38,526
This observation is incredibly powerful.

45
00:02:38,526 --> 00:02:41,006
It means that wherever you are in line,

46
00:02:41,006 --> 00:02:46,233
it’d be wise to trade your place for one
of the three spots in front of you.

47
00:02:46,233 --> 00:02:50,177
Let’s ignore which for now and think 
of them as a trio—

48
00:02:50,177 --> 00:02:54,958
this trio’s maximum value 
is better than this trio’s, and so on.

49
00:02:54,958 --> 00:02:57,356
Keep going and you’ll reach the front...

50
00:02:57,356 --> 00:03:01,938
These three spots must contain 
the extreme values—

51
00:03:01,938 --> 00:03:04,218
the best and worst probabilities—

52
00:03:04,218 --> 00:03:05,826
for the entire line.

53
00:03:05,826 --> 00:03:09,928
In other words,
they’re all we need to consider.

54
00:03:09,928 --> 00:03:11,248
Place 1 is bad.

55
00:03:11,248 --> 00:03:14,718
Head one would save you, and the other
two doom you forever.

56
00:03:14,718 --> 00:03:17,508
That’s just a 1 in 3 chance to escape.

57
00:03:17,508 --> 00:03:21,058
Place 2 is better: head two is great,

58
00:03:21,058 --> 00:03:26,652
head 3 is bad, and head 1 is ok
in that it gives you another chance.

59
00:03:26,652 --> 00:03:31,324
But place 3 is best, 
because head 3 saves you

60
00:03:31,324 --> 00:03:35,991
while heads 1 and 2 both
give you extra chances.

61
00:03:35,991 --> 00:03:39,331
If you did want to consider 
the exact probabilities,

62
00:03:39,331 --> 00:03:45,948
the odds of surviving in place 3 
are 16 out of 27, or close to 60%.

63
00:03:45,948 --> 00:03:51,616
The spots later in line tend to be very 
close to having a 50% chance of survival.

64
00:03:51,616 --> 00:03:52,936
Why 50%?

65
00:03:52,936 --> 00:03:57,204
Because every time Cerberus sends
one soul up to be reborn,

66
00:03:57,204 --> 00:04:01,571
he leaves 0, 1, or 2 souls
in the underworld.

67
00:04:01,571 --> 00:04:07,040
That averages out to one person
staying for each one who gets freed.

68
00:04:07,040 --> 00:04:11,105
But you can beat those odds handily
with what you now know.

69
00:04:11,105 --> 00:04:14,055
Hermes has places to be, and so do you.

70
00:04:14,055 --> 00:04:18,055
He rewards your insight by sneaking 
you into the third spot.

71
00:04:18,055 --> 00:04:22,405
And from there it’ll be just a short wait 
to learn your ultimate fate.