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The Way To Bet Angel Obregon
"The race is not always to the swift,
nor the battle to the strong, but
that's the way to bet" -- Ring Lardner
Evil John had wrecked the hydrofoil on a submerged clump
of cypress roots and rather than risk an argument I shot
him. An alligator with a groupie stare took him tenderly
by the leg and dragged him lovingly into deeper water. Evil
John took it all very badly. Very Badly. His problem was
that he lacked a sense of history.
I slung the flight bag full of cocaine and was 10 yards
into the cypress swamp when I meet this bone-white Cuban
kid wearing 15th century Spanish armor, only made of turtle
shells with armadillo hides for the elbow joints. He said
his name was Ponce de Leon which is believable since I can
understand him and the brand of Spanish they taught me in
highschool hadn't been spoken in 500 years.
On the table were 3 goblets (they're better at Faber's),
1 contained water from the fountain of youth and the
other 2 contained tasteless odorless poisons for which
there was no antidote And he put forth the following
proposition: I could drink from 1 (or more) of the
goblets or take my chances with the swamp.
I thought of getting him in a blindingly painful Aikido
lock I had learned in the advanced marketing course at
Wharton, but he had me covered so I said.
"Too simple. Suppose I choose a goblet and
then,before I drink you remove 1 goblet and
let change me my choice to the 1 remaining."
He knew it was a trick, but he didn't see how and
didn't want to admit it so, he said:
"So what?, the odds are still 2 to 1 against you
which ever goblet you pick."
"No, they're not. Since the odds are
2 to 1 that I choose the wrong goblet,
switching goblets improves my chances
of winning from 1/3 to 2/3."
Ponce de Leon, gallant old fellow that he is, tried
to shake it off like a punch. He's clearly upset.
"That doesn't make sense.
2
"Of course it does. Let's describe a typical
outcome of the game by a 4-tuple: (u,v,w,x)
where {u} is the goblet I choose, {v} is the
goblet you remove, {w} is the goblet I
switch to and {x} stands for {W} or {L}
meaning: whether I win or lose.
For example, the 4-tuple (1,2,3,L} stands for:
I chose goblet number 1, you remove goblet number
1(the one with water from the fountain of youth),
I switch to goblet number 3 (the only one left)
and I get killed because it's poison. The sample
space can be written
S = {(1,2,3,L), (1,3,2,L), (3,2,1, W) "
"Wait a minute."
"No, consider the entire sample space of all
possible outcomes: Let's say that your initial
choice isgoverned by the uniform probability
distribution:each of the doors has probability
= 1/3. Now go back and look at S.
You are initially going to choose door 2 with,
say, probability = 1/3. The only outcome in
with 3 in the 1st position is (2,3,1,W). So,
we must assign probability =1/3 to this outcome.
By a simple change of variable, I am going to
choose 3 with probability 1/3. and the outcome
in S corresponding to this sequence is
(3,2,1,W) and this outcome is also assigned
probability = 1/3.
But in S, if I choose 1, I lose and I can
lose in 2 distinct ways. Our reasoning
here is that the event you initially
choose goblet 1
E(1) = {(1,2,3,L), (1,3,2,L)
has probability 1/3. Without further
assumptions the probabilities of the
individual events (1,2,3,L) and (1,3,2,L)
are not uniquely determined. But for our
problem these probabilities are irrelevant
as we now see.
For let P(1,2,3,L) = a, P(1,3,2,L) = b
where a + b = 1/3. The event I am interested
in is E(I win) = {(2,3,1),(3,2,1)}
From the rules: P(I win)=1/3 + 1/3=2/3
And what's more, P(I lose) = 1/3
3
So this answers our question--according to our
assumptions about the game, switching choices
gives you probability 2/3 of winning and 1/3
of losing"
"Wait, there are 2 outcomes, you get killed or
you dont."
"Calculate it out, it isn't that complicated.
Consider the set of 4-tuples."
"No. If you choose the right goblet and switch,
you get killed. If you a wrong goblet, I remove
the right one and when you switch it has to be
the other wrong one because it's the only one
left and again you get killed. If you switch,
you get killed every time. Your probability
of success = 0."
"Well, yes. I had assumed that you removed a
goblet at random. I should have said that.
And, if you do, the probability drops to 2
times 1/6 = 1/3. Calculate it again."
He sharpened a flamingo feather, dipped it in berry juice
and started scratching calculations.
"It still doesn't work. If you choose 2 then
there are 2 outcomes, not 1: S ={(2,3, 1,W),
(2,1,3,L)}"
"Well, yes, you have to add the assumption
that the goblet you remove is of those
with poison in it."
"In that case, it's obvious!"
He had been absent mindedly sipping at goblet 3 and he
now fell over dead. That brought the odds down to even
money. It was too good to pass up. I grabbed the middle
goblet and gulped.
Leslie
I didn't know you went to Wharton.
David
I had no defense against my parents fantasies.
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