[Music]
maybe the fates got clumsy maybe
poseidon had one of his angry days
however it happened the underworld is
overcrowded
and zeus has ordered hades to let some
spirits out
hades arranges all the souls of the dead
in a line before
cerberus when one of his three heads
bites down on the soul in front of it
they'll get returned to the land of the
living anyone to the left
must get out of line and stay in hades
forever
and everyone else shuffles forward at
which point
cerberus will feed again each of the
dog's heads
has an equal chance of being the one to
bite each time
and no two ever bite simultaneously
unfortunately hades minions forgot to
tell you what was happening
and by the time you show up there are
only 99 souls
left in line hades looks furious and
drawing attention to yourself won't end
well
but suddenly time freezes and hermes
steps out of the shadows he tells you he
can instantly put you into the line
and no one will realize what happened
but he'll only grant his grace
to someone clever enough to take full
advantage of it
choose the best place in line and he'll
give you the spot
choose wrong and he'll leave you to rot
which spot should you pick
it's possible to calculate the exact
probability of going free
in all 100 spots but there's a much
simpler path to the solution
that requires surprisingly little
calculation
imagine being anywhere in line way up at
the front
one of the three heads will pick someone
at random and you'll move forward
one two or three spaces since each
is equally likely your chance of
survival from wherever you started
is the average of the chances from each
of the three
spaces in front of you and this is where
you can find
a huge shortcut averages
must be on or between the extremes of
what you're averaging
they can never be higher than the
highest value or
lower than the lowest so whatever your
chances of survival are
where you start one of the three places
in front of you
is at least as good and probably better
this observation is incredibly powerful
it means that wherever you are in line
it'd be wise to trade your place
for one of the three spots in front of
you let's ignore
which for now and think of them as a
trio
this trio's maximum value is better than
this trios
and so on keep going and you'll reach
the front
these three spots must contain the
extreme values
the best and worst probabilities for the
entire line
in other words they're all we need to
consider
place one is bad head one would save you
and the other two doom you forever
that's just a one in three chance to
escape place
2 is better head 2 is great
head 3 is bad and head 1 is ok
in that it gives you another chance but
place 3
is best because head 3 saves you
while heads 1 and 2 both give you
extra chances if you did want to
consider the exact probabilities
the odds of surviving in place 3 are 16
out of 27
or close to 60 percent the spots later
in line tend to be very close to having
a 50
chance of survival why 50 because
every time cerberus sends one soul up to
be reborn
he leaves 0 or two souls in the
underworld
that averages out to one person staying
for each one who gets freed
but you can beat those odds handily with
what you now know
hermes has places to be and so do you he
rewards your insight by sneaking you
into the third spot
and from there it'll be just a short
wait to learn your
ultimate fate looking for something to
challenge your mind while you wait
try out brilliant brilliant has more
than 60 math
science and computer science courses to
help you keep
learning year round and while you're at
it you'll find plenty of
interactive code writing challenges and
problems to solve
see for yourself at brilliant.org ted ad
plus the first 833 of you to use that
link
will receive 20 off the annual premium
subscription fee
[Music]
you