Re: Linford's Fallacy


"Herman Jurjus" <hjmotz@xxxxxxxxx> wrote in message
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Jon Slaughter wrote:
"Emmanuel Goldstein" <religionisacrutch@xxxxxxxxxxx> wrote in message
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"While it is true that flipping out any given sequence of heads/tails is
equally probable, it is not true that the probability of having all
heads is the same as the probability as having any mixture of heads and
tails. There are many more series of coin flips with a mixture of heads
or tails than there are series of coin flips with only heads.
Demonstrating with 5 coins, as you recommend, consider the following
list, where H and T are defined in the obvious way:

HHHHH
HHHHT
HHHTH
HHHTT
HHTHH
HHTHT
HHTTT
etc. all the way to
TTTTT

Now if you write all those out (please do) you'll fnd out there are
2^5=32 possibilities. However, only 1 of them is all heads (the first
one on the list.) So given that each of these series is equally
probable, there is a probability of 1/32 to get all heads. Of course
this is the same for any combination of heads and tails. However, the
probability of getting any arbitrary mixture of heads and tails is
1-2/32 or 93.75%. And the probability for a mixture converges to 1 as
the system gets larger and larger (this is the Law of Large Numbers.) To
see this, notice that everything on the list except for the first and
last entry is a mixture. So there are many more mixtures than there are
pure combinations."

I know the above is wrong. The question is, what exactly has gone wrong
here? I think it has to do with being able to extend the same argument
to any string. Can someone explain the problem?

Wrong? Why is it wrong? There is only 2 sequences, no matter what, of all
heads or all tails. If the string lenght is n then there are 2^n -2
strings that contain at least one opposite. There are only 2 that contain
all the same. Hence the probability is (2^n-2)/2^n -> 1

Yet, if we participate in a lottery, and the winning ticket turns out to
be '1 2 3 4 5 6 7', we would be very surprised, and even /strongly/
suspect foul play. But not so with '41 23 37 4 19 21 62'. How come?

(This seems to be more a matter of psychology than of mathematics.
... At least at first sight.)


This has nothing to do with what the OP quoted. The original statement is
mathematically sound. It has nothing to do with psychology. Mathematics is
not precipitated on peoples erroneous logic.

The lotto example is fallacious because the probability of getting a
sequential string is much lower than a random string. That is what people
are looking at. It is true that the probabiliy of any sequence is the same
it is not true that random sequences have the same probability as
"non-random" ones.

Hence people expect to see random sequences much more often(because it is
suppose to be randomly chosen). When such a sequence occurs that has such a
low probability(similarly with getting all heads on a flip) people rightly
question if it is due to fraud. Of course one can't answer such a question
mathematically without knowing more information. All a mathematician can say
is that it is possible that such a sequence can occur.

But if such a sequence came up several times then we can say something about
it(at least for all practical means).

Similarly if one was to get 41 23 37 4 19 21 62 10 times in a row. While it
is possible it is not probable.


for example, supposing there are 62 numbers then the chance of getting the
1..7 is 1 in 62^7. No one expects to see that in their life times. Hence
when it does happen it is supprising. We don't expect much from the others
even though technically they also have a 1 in 62^7. This is because there
are many more random looking sequences than non-random ones. Hence it is
much more probable to get a random looking sequence than not and hence it
doesn't bother us when we see them.

It's psychological because if it were purely mathematics then one wouldn't
question a string of 1 million heads in a row. It has a chance so... But for
practicaly everyone in the world would question the result. Why? Because the
chance of them being wrong is extremly small. Again, why? Becase there is
only 1 sequence of a million heads out of 2^(10^6). It's possible but highly
improbable and I'm sure some people would bet their life against it.

Similarly, the number of psuedo-random looking sequence in the lotto are
much higher than the number of non-psuedo random looking sequences. It is
psychological up to the point in determing if a sequence looks random. Do a
imbecile the sequence 17, 41, 33, 20, 8, 13, 59 looks random but it is
simply generated from floor(Zeta(2)*10^n)%62+1 or the factors from .

One can find any deterministic algorithm to express seemingly random
sequence of digits so that they are not really random looking then.


We can see the truth in this simply looking at all the lotto results. Is
there even one sequence that is sequential? I doubt it. Even though any
particular sequence has the same probability as every other we are choosing
from a sample space that can be categorized into areas of probability. We
see that the category of psuedo random looking sequences is much larger than
those of sequential sequences.

For logic with the lotto example is this: If we have a bag with 10^6 balls;
10^6-1 blue and 1 red. Then we would not be supprised if the red was chosen
on the first draw(or really any draw). We could group all the blue balls
into an equivalent problem with two balls... one blue and one red with the
red ball having probability of 1/10^6 of being chosen(using some method).

The distintion is exaclty that. We cannot distinguish the blue balls. If
each ball had a different color then we could and wouldn't think anything of
it if the red ball was chosen.

Similarly, while mathematically each sequence in the lotto example is
identical in terms of probability... they are not identical in how our minds
percieve them. We can easily remember sequential sequences but can't
remember random looking ones.


While how our minds create the different categories is not mathematical, the
fact that the categories exist and are, in and of themselves, a probability
space, is.

Remember, our brains work on determining probabilities and we see more
random like sequences because they look random to us... because we don't
have some relation for those sequences. If that sequence was your birthday
then it would be special and have some meaning to you and you would very
supprised if it came up. I guess we expect non-meaningful sequence to come
up more because we are not that intelligent. i.e., we see almost everything
as random and what little we relate to is so small that when we do see it we
are supprised.







.

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