

Subject:
Probability of opening to a page in a book
Category: Science > Math Asked by: gatecrasherga List Price: $8.00 
Posted:
19 Jul 2005 22:53 PDT
Expires: 18 Aug 2005 22:53 PDT Question ID: 545632 
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! 

Subject:
Re: Probability of opening to a page in a book
Answered By: tillga on 20 Jul 2005 00:00 PDT Rated: 
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 headsup or tailsup, 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. tillga Search strategy: I used the internal serach function of http://www.wikipedia.org with the search term "probability" 
gatecrasherga
rated this answer:
The initial answer was satisfactory. I appreciate the fact that tillga continued to defend the answer given and commented in the discussion in response to other comments. Thanks! 

Subject:
Re: Probability of opening to a page in a book
From: cartmanqbga on 20 Jul 2005 04:10 PDT 
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 runofthemill 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 beginninglevel 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 
Subject:
Re: Probability of opening to a page in a book
From: myoaringa on 20 Jul 2005 04:29 PDT 
"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 
Subject:
Re: Probability of opening to a page in a book
From: justaskscottga on 20 Jul 2005 05:18 PDT 
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. 
Subject:
Re: Probability of opening to a page in a book
From: tillga on 20 Jul 2005 06:58 PDT 
I disaggree with the comments by myoaringa and justaskscottga. 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 tillga 
Subject:
Re: Probability of opening to a page in a book
From: journalistga on 20 Jul 2005 08:11 PDT 
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, journalistga 
Subject:
Re: Probability of opening to a page in a book
From: mikewaga on 20 Jul 2005 08:27 PDT 
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. 
Subject:
Re: Probability of opening to a page in a book
From: myoaringa on 20 Jul 2005 09:40 PDT 
mikewaga, But each day another "piece of information" is the target. Tillga and Journalistga, 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 
Subject:
Re: Probability of opening to a page in a book
From: eppyga on 20 Jul 2005 14:13 PDT 
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. 
Subject:
Additional details
From: gatecrasherga on 20 Jul 2005 23:37 PDT 
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 
Subject:
Re: Probability of opening to a page in a book
From: myoaringa on 21 Jul 2005 00:27 PDT 
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 
Subject:
Re: Probability of opening to a page in a book
From: tillga on 21 Jul 2005 01:04 PDT 
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. tillga 
Subject:
Re: Probability of opening to a page in a book
From: gatecrasherga on 22 Jul 2005 00:00 PDT 
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 5060 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... 
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