bjbjLULU Episode 37: Game Show Theory Hello
Internet. Welcome to Game Theory, or should
I say Game Show Theory. Because gaming isn
t just limited to consoles and controllers,
and while, yes, it may be interesting to know
that Jynx isn t racist, chances are it won
t help you win A BRAND NEW CAR! So this week,
we re entering a world of fabulous prizes
and gaudy suits, a world of Bobs, from Barker
to Eubanks, with more Rods and Winks and Dicks
than you can shake a skinny microphone at.
So start pricing up your turtle wax and Rice-a-Roni,
Joe Millionaires, because this week I m teaching
you how to Press Your Luck without becoming
the Weakest Link. In other words, how knowing
simple probability can help you win fabulous
prizes on daytime television. In a few weeks,
I ll be appearing in the studio audience of
Let s Make a Deal, a classic game show from
the 1960s that was revived in 2009 when the
entertainment industry officially ran out
of creativity. Now, I won t know if I m actually
going to be a contestant until I get there,
but it pays to come in prepared. Oh, and if
you re wondering why people are dressed like
Chef Boyardee, remember that this is TV, so
if they re going to give you something, gosh
darn it they ve got to humiliate you in front
of millions of people watching from home first!
So, if you ve never seen the show before,
the premise is that the host walks around
the studio audience giving away prizes: money,
mystery boxes, whatever, and then offers contestants
the choice to either keep what they have or
trade it for something hidden behind a curtain.
One tends to be a nice prize, while the other
is an ostrich, an adult male in an oversized
crib, or some other creature you don t really
want to keep in your back yard. These joke
prizes are called Zonks. At the end of the
game, the top winners are presented with three
doors. Hidden behind one is the big prize
of the day, while the other two hide more
Zonks. So, let s assume I go to the show and
get to select one of the three final doors.
Behind one is a brand new puce-green kitchen
set straight out of the Johnson-era while
behind the other two are donkeys or some other
beast of burden. I choose a door, say door
1, at which point the host opens one of the
doors I didn t pick to reveal a donkey. He
then says that I can change my door if I want
to. Assuming I want a vomit-colored kitchen
set, should I switch or stay? Does it even
make a difference? It seems like it shouldn
t matter, right? With only two doors left,
my odds seem to be 50-50, one door hiding
a hideous Frigidair and the other holding
my future pet burro, Bruce. But think again.
At the beginning of the game, I had a 1/3
chance of choosing correctly and a 2/3 chance
of choosing a Zonk, at which point the host
always opens a door with a donkey. If I chose
the car initially and then switch, well I
get my ass handed to me, literally. But if
I chose a donkey in the first round and then
switch, I m guaranteed to win, since the other
donkey has already been revealed. Thus, with
my first choice, I m hoping to choose a donkey.
Since my odds of choosing a donkey at the
beginning of the game was 2/3, by switching,
I now have a 2/3 chance of walking home with
a refrigerator that ll make the EPA tremble
in fear. In other words, choosing to switch
always maximizes your chance of winning. If
you re still confused, think about it with
100 doors: 99 donkeys and one kitchen set.
You pick a door. Obviously, you have a 1/100
chance of choosing correctly, which isn t
very likely. But then the host, who knows
exactly where the prize is, opens 98 doors
with donkeys before offering you the chance
to switch. The odds that the kitchen set is
behind your door is still 1/100, but because
the host knowingly cut down the field to one
other specially-selected door, the odds are
99/100 that by switching, you ll win. It may
seem like sticking with your door is the right
choice, especially now that you re so close
to winning, but in actuality, you re no closer
than when you originally started. This situation
is actually a famous probability problem named
the Monty Hall Problem, named after Let s
Make a Deal s first and most famous host,
which leaves me with just one more question:
How was this color ever in fashion? The 60s
were a weird time, my friends. But no game
show marathon would be complete without a
trip to the granddaddy of daytime gamery,
the highlight of every elementary student
s sickday, the Price is Right, specifically,
the game that everyone loves to see PLINKO!
102 pegs, 5 chips, $50,000 on the line. Drop
a chip, win the amount of the slot it lands
in. Elegant, addicting, and impossible as
fu**. You see, the game has been a part of
the show since 1983, and yet, for as many
times as it s been played, no one has ever
earned the maximum prize. How could they considering
it s a game where you randomly drop tokens
down a pachinko board? So is there any way
to have the odds be ever in your favor? OF
COURSE, but only with the help of the sixth,
and sadly forgotten Planeteer, Math! (Earth,
fire, wind, water, heart) MATH! Oh come on,
it s no just as lame as Heart power! Here
s a diagram of the board. Though there are
a total of 102 pegs on the board, a chip will
only hit twelve on its way to the bottom.
Assuming that at each peg, the chip is equally
likely to go either left or right, where should
a contestant drop their chip to have the best
chance of landing in $10,000? Let s start
with slot 5, right in the middle and directly
above the $10,000 space. If the chip is going
to earn us the most money, it ll need to travel
left six times and right six times, in no
particular order. To figure out the probability
of this happening, we can use what s known
as the binomial distribution. This is the
formula, which is one of those intimidating
math formulas that aren t so bad if you work
through them slowly. This top number represents
the number of trials, or in our case 12 for
the number of pegs our chip will hit. This
bottom number is the number of times we need
it to go left. Since we re assuming the chip
has an equal likelihood of going left or right,
the probability is for each, which goes here
and here. And finally, this is how many times
we want it to go left, and this is how many
times we want it to go right. Through some
boring calculations, we find that the probability
of it landing in the middle if dropped in
the middle is about 22.56%, or about 1/5 of
the time. Not bad, but can we do better? Take
a look at what happens if you try dropping
it in either slot 4. We would need it to go
left five times and right seven. Plugging
it into our equation gives us about 19.34%,
so significantly less. Slot six would be the
same since the board is symmetrical. And the
odds continue to decrease as we move away
from the center. In short, the Plinko board
is a type of normal distribution, or bell
curve, where more things wind up towards the
center than either extreme. So, our best strategy
is to drop it into the center slot. That said,
what are the odds we walk away with the grand
prize of $50,000? When determining the odds
of multiple events all happening at the same
time, you multiply their probabilities of
happening together. Thus, getting all 5 chips
into the $10,000 slot means 22.56 to the fifth
power or .0584%. It s no wonder no one has
ever officially won the game! With odds that
low, you re more likely to see Michael Bay
direct an Oscar-winning art film. So, today
s episode may not guarantee you a win, but
the next time you re in an audience and told
to Come on down, remember your math, loyal
theorists, and you may just be lucky enough
to go home with a grill shaped like a large
green egg. Now if you ll excuse me, I have
a strange urge to go spay and neuter my cat.
But hey, that s just a theory. A game show
theory. Thanks for watching! h}p* h}p* gd?d
h'u- Episode 37: Game Show Theory Matthew
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