I don't know anything about probability, so I'll just explain what I
am looking for as good as I can:

Let's say there is a book with 1637 pages. Let's say there is one
individual person, and that person is looking for a specific piece of
information within the book. (the information would be on only one
page) What are the chances that the individual person would open the
book exactly to the page needed, *without flipping any pages* and
*without intentionally trying to do this.* In other words, what are
the odds that the person would be lucky and immediately open the book
to the page needed with no intention of doing so?

If this person were to look up one different piece of information
every other day in the same 1637 page book, how often would it be
considered "normal" for the unintentional "lucky page opening" to
happen?

The reason I ask this question is that I feel that I experience the
phenomena described at a rate higher than would be considered
"reasonable" and I am curious as to how my experiences compare with
actual probability.

As I said, I don't know anything about probability, so please try to
explain it as simple as you can. Also feel free to ask for
clarification.

Thank you!

The answer is rather simple:

The probability for opening one of the pages is
1 / 1637 = 0.000610873549

Compare your problem to the odds for the tossed coin:

"Representation and interpretation of probability values
The probability of an event is generally represented as a real number
between 0 and 1, inclusive. An impossible event has a probability of
exactly 0, and a certain event has a probability of 1, but the
converses are not always true: probability 0 events are not always
impossible, nor probability 1 events certain. The rather subtle
distinction between "certain" and "probability 1" is treated at
greater length in the article on "almost surely".

Most probabilities that occur in practice are numbers between 0 and 1,
indicating the event's position on the continuum between impossibility
and certainty. The closer an event's probability is to 1, the more
likely it is to occur.

For example, if two mutually exclusive events are assumed equally
probable, such as a flipped coin landing heads-up or tails-up, we can
express the probability of each event as "1 in 2", or, equivalently,
"50%" or "1/2".
Odds a:b for some event are equivalent to probability a/(a+b). For
example, 1:1 odds are equivalent to probability 1/2, and 3:2 odds are
equivalent to probability 3/5."
from Wikipedia - the free encyclopedia
( http://en.wikipedia.org/wiki/Probability )

In your case the probability of one event (opening the specific page)
must be divided through the one for all events.

Mathematically there´s no explanation for the phenomena you seem to
have found. You must see hoewever that these statistical rules do not
apply for a low number of tests (=book openings).

Again let us compare your problem to the toin coss:

"To give a mathematical meaning to probability, consider flipping a
"fair" coin. Intuitively, the probability that heads will come up on
any given coin toss is "obviously" 50%; but this statement alone lacks
mathematical rigor - certainly, while we might expect that flipping
such a coin 10 times will yield 5 heads and 5 tails, there is no
guarantee that this will occur; it is possible for example to flip 10
heads in a row. What then does the number "50%" mean in this context?
One approach is to use the law of large numbers. In this case, we
assume that we can perform any number of coin flips, with each coin
flip being independent - that is to say, the outcome of each coin flip
is unaffected by previous coin flips. If we perform N trials (coin
flips), and let NH be the number of times the coin lands heads, then
we can, for any N, consider the
ratio NH/N.
As N gets larger and larger, we expect that in our example the ratio
NH/N will get closer and closer to 1/2"
(from same source as above)

So opening the book only a few times will not bring about the exact
probabilty given above but you will approach the given number when
opening the book very often. This might be the explanation to your
problem.


I hope this answers your problem.

till-ga


Search strategy:
I used the internal serach function of http://www.wikipedia.org with
the search term "probability"

The initial answer was satisfactory. I appreciate the fact that
till-ga continued to defend the answer given and commented in the
discussion in response to other comments. Thanks!

wow, i think that is a poor answer.

there are many other factors that could be accounted for by your
question that make the question a lot more fun that the
run-of-the-mill response given.

First, it depends what type of book and what type of question you
have. For example, if you are looking at a book describing "Economics"
(a very broad book indeed) and your question is an beginning-level
question, the answer would be more likely to be found in the beginning
of the book, right? It seems easier to presume such a reality. Even
though the subject may believe s/he is randomly opening the book, it
might be that they subconciously look closer to that area that is more
prone to give them the answer they seek.

Further, most people don't open a book to the first 20 percent or last
20 percent of a book. The book binding is created such that when you
randomly open it, it usually falls somewhere close to middle. Perhaps
your the questions that you 'randomly' ask are more likely to fall
into the middle of a book, and that is why you experience this
surprising result of always finding what you need instantly.

but, then again, till did give a suitable answer

"The reason I ask this question is that I feel that I experience the
phenomena described at a rate higher than would be considered
"reasonable" and I am curious as to how my experiences compare with
actual probability."

There could be a few things that influence your beating the statistics.
I have difficulty imagining a book with purely randomly sorted  items
of information, but which are also items that I can choose from some
list, because I have to know before I touch the book which item will
be THE one on that day.

If the items are sorted in ANY way, and I am doing the experiment by
hand as you describe, I suspect that my fingers would not be just
randomly opening the book, even if this was limited to unconsciously
eliminating a few of the first and last pages of the book.
This could be gotten around by have a computer program randomly pick a
page by number.

Another factor is that the experiment with the book always provides
two pages on which the item could be.  Since you did not address this,
I expect that you had not considered it.  This doubles your odds, of
course, so this alone could account for and justify your impression
that you are beating the statistics, but in this case the statistics
used have been wrong, because the "strike" factor is not 1637 but 818
and a half.

Okay?  Myoarin

I concur with myoarin's comment about the need to consider that the
reader will open randomly to two pages, rather than one.  The reader
might look at only one of those pages; but strictly speaking, the
reader will open to two pages.  Assuming that the book has 1637 pages
of content, with no material before, after, or in between (such as
title page, index, or pages solely with chapter numbers), only one
page -- the first or the last -- will be by itself.  The odds of
opening to any page at random, as myaorin indicates, should be 1 in
818.5.

I disaggree with the comments by myoarin-ga and justaskscott-ga.
Opening a page in the sense given here must be considered as opening
one page page not two.
The point is that the specific piece of information will be available
on just ONE page. So if you open the book you will have to decide if
you will read the left or right page. But this will not change the
odds.
In formula:
Odds for opening a pair of pages: 1/818.5
Odds for reading left or right page: 1/2
total = 1/828.5 * 1/2 = 1/1637

till-ga

I agree with Till because it would be very difficult for a person to
read two pages at a time.  Just generally opening a bok would reveal
two pages, however only one page would reveal the info (even if the
info begins at the bottom of one page and travels to the other, the
average person would first see only one page).

Regarding the actual location of material in one look (if this is what
initially happened): If this is a book the customer has read in the
past, chances are the page number or location of the info is within
the reader's memory, even if he/she can't consciously recall it.  So,
it's possible the unconscious recalls the exact page, assisting the
conscious act of opening to that page.

Best regards,
journalist-ga

I agree that the probability of opening the book at random, and
finding the page you want is theoretically 1/1637 but, as I understand
it there is a second part to the question, namely 'the book seems to
open at this page more frequently than expected.'
This could be due to a couple of factors. First, now you know where
the page is, you subconciously tend to open the book at that point
again. Second, the book might really open more often at that page. If
it contains a subject that many people have looked up, then the
binding might become damaged enough to favor that page being opened.

mikewa-ga,
But each day another "piece of information" is the target.

Till-ga and Journalist-ga,
We have to wait and see what Gatecrasher says.  I made the assumption
that he would have mentioned that he was limiting his first glance to
just one page if that were the case.  Since he/she did not  (and I
probably wouldn't have either) I  thought this could be significant.

Let's see ...
Myoarin

A couple more points. If the same information had been looked up
several times before, the spine of the book would tear slightly over
time so as to make it more likely to open to the same page.

Secondly, the question included a comment about "experiencing" the
phenonema more that what was perceived as "reasonable". When you deal
with personal memories,you are moving from statistics into psycholgoy.
In a nutshell, we rememmber the unusual and unlikely events more than
the mundane. As such, you are very likely to remember everytime you
"fluked" the book opening to the correct page, but very unlikely to
remember every time you've opened the book to the wrong page. This may
create a *perception* of the unlikely event occuring more often in
comparison to expected event than is actually the case.

The following details were realized as I read the answers/comments
given by everyone. I can see now that there are some things that I had
not considered which substantially change the results. I also have my
own opinions on some of the comments given:

First and foremost, I am embarassed to say that I had not indicated
the fact that the book is arranged alphabetically. (I know that was
foolish to miss). Though I am not an expert on the subject, my opinion
is that mathematically the answer is either 1/1637 OR 1/818.5 (not
sure which), but theoretically it is more likely due to the
alphabetical organization of the book.

I agree that it is extremely likely that the incidence of the
phenomena is higher due to the fact that subconsciously my mind is
selecting a general portion of the book that is most likely to contain
the desired data. As mentioned, usually the first and last portions of
a book are less prone to be opened unintentionally. This is actually
both a defense and a support to the situation. The odds that I open
the book to something in the first or last section of a book are
reduced due to the fact that these sections are least opened to when
not intentional. However, the odds of opening to a page accurately yet
unintentionally at the beginning or end of the book are actually
increased due to the fact that (because the book is alphabetical) the
subconsciousness would encroach toward consciousness further as the
desired information gets closer to the front or back of the book.

To answer the question of which page I look at: When I open the book
(which would inherently reveal 2 pages) I generally observe the
letters and move up or down in the alphabet until I find the data on
one page or the other. It's hard to say if this really divides the
probability in half or not. Seems like it does, because for every time
I open the book, I have 2 chances of having the correct page revealed.

Despite all this, I would like to add something that may come of interest:

This phenomena does occur quite often for me in OTHER books or
magazines, some of which are randomly organized. I am not sure how the
odds change under those circumstances.

Regarding the law of large numbers, would it be true that the
probability of my phenomena are decreased if they happen more often
during a smaller amount of openings of the book?

One thing that I am extremely interested in is some of YOUR
experiences with this phenomena. Have any of you experienced this
phenomena? How often has it happened/how many times in your life? What
do you feel when it happens?

Mathematically the following is not possible to answer, but in your
opinion, what ratio of this phenomena to the # of page openings seems
"reasonable to you." This is meant to be a subjective question, please
comment.

Thanks for everyone's help, and please continue the discussion!

Gatecrasher

Gatecrasher,
Thank you for your detailed reply.  Muchly appreciated.

Personally, I cannot remember once opening a book to the page I
wanted, and I have  - lets see - five dictionaries here and use them a
lot, plus frequent references to two encyclopediae.

You must be psychic  ;-)

Myoarin

My personal experience is that I have never observed the phenomenon
you describe. In my humble opinion this must be coincidence.
Or maybe psychology plays a role here: maybe the "hit" you get by
finding the info is remembered better than a miss.

till-ga

I performed some experiments today. The unfortunate thing about this
phenomena is that it is nearly impossible to test scientifically, due
to the "unintentional" and "subconscious" aspects. However, I thought
it would be interesting to try a few things anyway. I held the same
alphabetical book, without actually placing my fingers on the covers
directly, and without looking at the book, and had my dad randomly
name a word. Then I quickly, thinking as little as possible, just
opened the book as fast as I could. 2/3 times I opened the book at
least to the beginning letter(s) of the word he mentioned, without
looking at the book or evaluating too much. Then, I had him hold the
book in front of me in the position that he would look at it (making
the book upside down to me). I closed my eyes, held my hands above the
book without touching it, and had him randomly think of then mention a
word. As soon as he said the word, I put my hands down and opened the
book, trying not to touch the covers. I opened it to the beginning
letter of the word he selected. Finally, I held the book in hand,
tried to decide on a word...I saw my cat and chose his name (Rusty).
Immediately upon deciding to look up "Rusty" I opened the book,
without looking at it, and when the book was open, I viewed the
revealed pages. The word Rusty, IF it would have been in the book,
would have appeared on the exact page to which I both opened to and
first looked at!

I'll admit that I gave my dad a try. He held the book, without
looking, and I selected a word from my mind. He opened the book and
was three pages away from the word. I never expected him to be that
close.

Unfortunately as mentioned before, this was not a valid experiment due
to the intentionality and more conscious state of mind under which we
did the test.

I agree that this phenomena could theoretically be a coincidence. I
also agree that it is true that we tend to remember the "unusual" more
than the usual. However, I disagree that these two statements fully
explain my experiences. Till, please do not take offense to the
following: I find it interesting that you would indicate that,
although you have *never* experienced or observed something strange
like this, and knowing it happens very frequently to someone, that you
would still explain it away as being purely coincidence. I wish that I
could survey a large group of people, so I could get a better idea of
how often this phenomena really happens in general. It seems to me
that not many people have experienced this (ever). Even myoarin and
yourself have indicated that you have never observed/experienced this
phenomena for yourselves. I'm 25 yrs old and I'd say in my lifetime
(this is an estimate) this phenomena has happened to me about 50-60
times, possibly much more, but I wanted to undershoot the amount due
to the fact that the longer ago it happened, the less I can remember
the frequency. But I'm pretty sure that 60 or more might be accurate.
Maybe this really is a common thing that people experience but just
don't talk about. Or maybe I am just the statistical anomaly, similar
to landing a flipped coin the same side up 20 times in a row or
something. Or maybe, just maybe, if you aren't completely locked into
science and math, it might be something beyond what we know...