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Can you solve the riddle and escape Hades? - Dan Finkel

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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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    Pause the video to figure it out yourself.
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    Answer in 3
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    Answer in 2
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    Answer in 1
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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.
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    In other words,
    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.
Title:
Can you solve the riddle and escape Hades? - Dan Finkel
Speaker:
Dan Finkel
Description:

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Video Language:
English
Team:
TED
Project:
TED-Ed
Duration:
04:24

English subtitles

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