>Suppose you tossed a fair coin 64 times, for each throw recording
>a 1 for heads and a 0 for tails. Would either of the following two
>sequences be more surprising?
>
>(A) 1010101010101010101010101010101010101010101010101010101010101010
>
>(B) 1110010001010001110000000000011101110101101110100101000111111010
>
>For greater impact, I should have generated 99 sequences by tossing
>a fair coin, put sequence A in somewhere and then said: "only one of
>the following sequences was not generated by tossing a coin, can you
>find it?
>
>The person I was talking to was honest enough to admit that he would be
>very surprised to see something like (A), but went on to say that,
>given what he knows about probability, he shouldn't be surprised.
>So close but yet so far, the reason that one could pick (A) out of
>99 random sequences and could find the one computer terminal out of
>a hundred (or a thousand or a million or a trillion) that has the
>strange attractor is because of the pattern. Random processes are
>expected to give random results. Very very occasionally they may not,
>but this is so improbable that it can be discounted. This is well
>known and can be proven, but I'm going to skip those details here.
>
>I dare say that very few natural laws would have ever been discovered
>if scientists actually employed the falacy mentioned above in practice.
The article in New Scientist which I recommended to the group the other day
includes the following:
"Our uneasy attitude toward randomness is probably to do with the human
penchant for spotting patterns. The brain has an architecture ideal for
picking out a person in a crowd, or linking together disparate events --
abilities that have obvious evolutionary advantages[I'm not quoting this
because I agree -- only to establish context - weh]. 'Humans want
order--and they will impose order even when it is not there,' says
Professor Norman Ginsburg, emeritus professor of psychology at McMaster
University, Ontario, who has made a study of how well humans simulate
randomness.
This love of order can be a severe handicap when you're dealing with random
phenomena. Take the apparently simple task of writing down a sequence of
100 'random numbers'. In research published in the Journal _Perceptual and
Motor Skills_, Ginsburg reported that volunteers had serious problems
making their numbers genuinely random. In particular they tended to avoid
repeating numbers and having sequences like 15, 16, 17. They also disliked
using numbers again until all the others had been 'given a go'.
But true randomness has no memory of what went before, and it is entirely
possible for small samples of random numbers to show fleeting bursts of
apparent order. Where people go wrong, it seems, is in thinking that such
snapshots of randomness have the characteristic lack of order that becomes
apparent only in the very long run... Statistics from Britain's National
Lottery suport this: in those weeks for which numbers drawn contain a
consecutive pair, most had no jackpot winner. By contrast, in one draw
where the numbers were separated from each other by at least three other
numbers, no fewer than 133 people shared the jackpot"
The point is that no sequence is inherently random or nonrandom. The
sequence Brian included could have come from tosses of a fair coin. The
article points out that these apparent patterns are typically "fleeting"
and that the lack of order typical of randomness typically requires long
strings of numbers. But how long is long? 100? 1000? 10^19?. Depending
on what tests you use and how persnickety you are, your mileage will vary.
Still, Brian makes an excellent point in reminding us that a quite a number
of scientific discoveries have begun with someone noting an anomaly in som
data and _not_ attributing it to randomness. A good example might be the
discovery of Uranus. Anomalies were noted (I believe) in the orbit of
Neptune, and one possible explanation was an undiscovered planet between
Saturn and Neptune. My understanding is that by doing some perturbation
analysis, astronomers were able to determine where to look for Uranus.
The moral of the story is that patterns that don't "look" random can be
random nevertheless, but we are likely to be the losers if we cavaleirly
assign all instances of unexpected pattern to the perverseness of short
random sequences.
Bill Hamilton | Vehicle Systems Research
GM R&D Center | Warren, MI 48090-9055
810 986 1474 (voice) | 810 986 3003 (FAX)