>Brian wrote:
>
>"1. HHHHTTT
>
>2. THHTHTT
>
>3. TTTTTTT
>
>On which would you bet?"
>
>True -- that all are equally probable. But there is one more piece of
>information
>you gave us -- you said YOU HAD ACTUALLY TOSSED the coins.
>
>So since #2 is more typical, and since it is more likely you tossed
>something
>typical than not -- I'd bet on #2.
>
>What am I missing?
>
Hello Burgy. No, you are not missing anything.
For those who may have missed it or don't remember,
the quote above is from a wager that I found in a book
by Piattelli-Palmarini. The quote was given because it
closely relates to probability arguments one often sees
in evolution/creation discussions. To refresh our memories,
here's the bet:
===============================================
I have just tossed a coin 7 times, and I ask you, who
have not seen the result, to guess which of the three
sequences below represents the sequence of my results.
I guarantee that one of the sequences is genuine. If
you don't get it right, you lose 10 dollars; if you
win, you get 30. H stands for heads, and T for Tails.
1. HHHHTTT
2. THHTHTT
3. TTTTTTT
On which would you bet? Let's think for a moment before
going on.
Experiments with a great many subjects have shown that
the bets will be placed in the following order:2,1,3.
The preference for the second sequence is very strong.
But probability theory tells us that in seven tosses of
a coin the probabilities are totally even, and we rationally
should be quite indifferent to which of the three sequences
we choose. The person who chooses 2 is prey to one of the
most common cognitive illusions; she mistakes the most
<typical> for the most <probable>.
-- Piattelli-Palmarini, <Inevitable Illusions: How Mistakes of
Reason Rule Our Minds>, John Wiley & Sons, 1994, p. 49-50.
====================================================
So, both Burgy and myself seem to have fallen prey to a "common
cognitive illusion", :) mistaking most typical for most probable.
What does the author have in mind here? OK, clearly the
probability of any specific sequence is the same, so if one
had to guess what a result would be in advance, perhaps he
has a point. But that isn't the way the bet works. So, why
isn't the most typical the most probable? Perhaps the problem
is (1) that one is stuck on the limitations of probability theory,
having to deal only with specific outcomes and (2) the idea of
"typical" is not very precisely defined. i.e., suppose we had to
defend our conclusion to a skeptic. They say, "OK smart guy,
prove that a typical result is the most probable outcome
from a single trial". If all one has is probability theory then
one seems to be stuck with answers like "isn't it obvious?",
or, as Rafiki said to Simba "Look haaarder" ;-).
In a recent post I mentioned results that have been developed
only in the last thirty years or so (algorithmic information
theory, AIT) which should lay this question to rest. In AIT
one *is* able to precisely define what "typical" means and
show that *any* result from a coin tossing experiment that
contains *any* pattern is very highly unlikely.
But suppose one doesn't know this result. What's the proper
way to settle the dispute? Is it through unwavering faith in
probability theory with condescending remarks about
cognitive illusions? No, one should do an experiment. Repeat
the wager many times and see who emerges with the most
money. Now, its very obvious that Piattelli-Palmarini didn't
actually perform this experiment. First of all, based upon the
results from AIT he would clearly lose in the long run. Secondly,
he really botched the odds. I could just guess randomly and
still win money in the long run.
I find this to be enormously interesting and entertaining
considering the title of the book <Inevitable Illusions: How
Mistakes of Reason Rule Our Minds>.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"It appears to me that this author is asking
much less than what you are refusing to answer"
-- Galileo (as Simplicio in _The Dialogue_)