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Maybe the fates got clumsy.

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Maybe Poseidon had one of his angry days.

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However it happened, the
underworld is overcrowded,

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and Zeus has ordered Hades
to let some spirits out.

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Hades arranges all the souls of the
dead in a line before Cerberus.

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When one of his three heads bites 
down on the soul in front of it,

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they’ll get returned to the 
land of the living.

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Anyone to the left must get out of
line and stay in Hades forever.

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And everyone else shuffles forward,
at which point Cerberus will feed again.

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Each of the dog’s heads has an equal
chance of being the one to bite each time,

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and no two ever bite simultaneously.

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Unfortunately, Hades’ minions forgot
to tell you what was happening,

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and by the time you show up there are
only 99 souls left in line.

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Hades looks furious and drawing attention
to yourself won’t end well.

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But suddenly, time freezes, and Hermes 
steps out of the shadows.

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He tells you he can instantly put you into
the line,

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and no one will realize what happened.

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But he’ll only grant his grace to someone
clever enough

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to take full advantage of it.

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Choose the best place in line and 
he’ll give you the spot.

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Choose wrong, and he’ll leave you to rot.

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Which spot should you pick?

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It’s possible to calculate the exact
probability of going free

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in all 100 spots.

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But there’s a much simpler path to
the solution

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that requires surprisingly
little calculation.

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Imagine being anywhere in line.

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Way up at the front, one of the three
heads will pick someone at random,

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and you’ll move forward 1, 2, or 3 spaces.

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Since each is equally likely,

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your chance of survival from wherever
you started

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is the average of the chances from
each of the three spaces in front of you.

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And this is where you can
find a huge shortcut.

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Averages must be on or between the 
extremes of what you’re averaging––

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they can never be higher than the
highest value or lower than the lowest.

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So whatever your chances of survival
are where you start,

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one of the three places in front 
of you is at least as good,

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and probably better.

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This observation is incredibly powerful.

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It means that wherever you are in line,

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it’d be wise to trade your place for one
of the three spots in front of you.

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Let’s ignore which for now and think of
them as a trio––

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this trio’s maximum value is better than
this trio’s, and so on.

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Keep going and you’ll reach the front...

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These three spots must contain the
extreme values––

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the best and worst probabilities––

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for the entire line. In other words,

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they’re all we need to consider.

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Place 1 is bad.

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Head one would save you, and the other
two doom you forever.

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That’s just a 1 in 3 chance to escape.

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Place 2 is better: head two is great,

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head 3 is bad, and head 1 is ok
in that it gives you another chance.

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But place 3 is best, because head 
3 saves you

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while heads 1 and 2 both
give you extra chances.

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If you did want to consider the exact 
probabilities,

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the odds of surviving in place 3 are 16
out of 27, or close to 60%.

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The spots later in line tend to be very 
close to having a 50% chance of survival.

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Why 50%?

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Because every time Cerberus sends
one soul up to be reborn,

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he leaves 0, 1, or 2 souls
in the underworld.

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That averages out to one person
staying for each one who gets freed.

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But you can beat those odds handily
with what you now know.

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Hermes has places to be, and so do you.

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He rewards your insight by sneaking 
you into the third spot.

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And from there it’ll be just a short
wait to learn your ultimate fate.