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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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