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Math Help - Interesting Probability about Shakespeare Prose

  1. #1
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    Interesting Probability about Shakespeare Prose

    If we shadows have offended,
    Think but this, and all is mended,
    That you have but slumber’d here
    While these visions did appear.
    And this weak and idle theme,
    No more yielding but a dream,
    Gentles, do not reprehend:
    if you pardon, we will mend:
    And, as I am an honest Puck,
    If we have unearned luck
    Now to ’scape the serpent’s tongue,
    We will make amends ere long;
    Else the Puck a liar call;
    So, good night unto you all.
    Give me your hands, if we be friends,
    And Robin shall restore amends.

    Pick any word in the first few lines and do the following: Let’s say your choice is the
    word “shadows”. This word has seven letters. The seventh word following “shadows”
    is “all”. This word has three letters. The third word following “all” is “That”. This
    word has four letters, etc. proceed until you come across the word “restore” in the last line, from where you cannot move any further. Why is it that no matter which word you choose in the first few lines, you always end up with the same word, “restore”. In fact, even if you had started somewhere in the middle of A Midsummer Nights Dream you would have wound up with the same word. And, even stranger, every play by Shakespeare contains a special word like “restore”. Are these words secret messages left by Shakespeare, is all this coincidence, or is there another easy explanation?


    Attempt at solution


    Not knowing how to exactly approach the problem, this is how far i got

    I first calculated the expected word length (number of letters/ number of words) = 371/94 = 3.95 (approx 4)

    Then i found the probability of landing on the 4 letter words (25 four letter words/ 94) = 0.266.

    Using the expected mean of 3.95 i solved for n => mean = n . p => n = 14.85

    so i believe n = the expected number of trials before the average word length reaches 4.

    I am stuck at this point. Am i even on the right track?

    thanks for your help in advance
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  2. #2
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    The way the problem is worded, I would think they are looking for a deterministic solution. Of course, given that you land on "restore" in this section if you start from the first few lines it is trivial that you could start anywhere in Midsummer Night's Dream, since you will eventually land on a word that is somewhere in the first few lines of this selection.

    This may not be what they are going for, but there are probably patches of words that are unavoidable that all converge to the same word. From then on out, the path you take is determined. Notice that if you start later on in the poem you don't necessarily end up hitting "restore." It probably becomes a statistical inevitability that such patches exist the longer your work is.

    Just as an example, if I had the phrase "But, am I?" in a work and I knew that I would hit just one of those three then I would know for sure what path would be followed after those words.
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  3. #3
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    Let me know if this sounds right,

    There are 7 words in the second last line that leads to the word restore, as well as the word "and", thus any words that have 8 or greater than 9 letters will not fall onto the word restore,

    Therefore, say if there was a word just before the second last line that had 8 letters or more than 9 than i would miss the word restore,
    so if i find the probability of there being a word with 8 or greater than 9 letters (approximately 6 % of words in this passage) than i can conclude that the probability of landing on one of the words that leads to restore is approximately 94%.

    does that resemble some sort of a proof?
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    Well, starting with the 4th-to-last line there are 17 consecutive words that lead to restore. Given that no word in the passage is greater than 17 letters long, you are going to go to "restore" no matter if you start from any word before the 17. That gets the job done for this particular poem but doesn't give you any real insight (I don't think I would talk about probabilities on a fixed passage since the words of the poem are presumably not random anymore).

    I think the main point of this, though, is that the book you choose is completely irrelevant. Things like this are inevitable. If I do the same thing with my Linear Models book, I hit the word "comment" no matter where I start in the first paragraph.
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    Okay thanks for your help so far,

    I am really stuck on incorporating the probability into this question. I cannot seem to find the mathematical reasoning. Although maybe its because its 1:30 in the morning :P
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  6. #6
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    Could I possibly use the mean of 4 letters per word, and show that this will converge onto the same path leading to restore. Which would also imply that the longer the paragraph is , the more probability that you will fall onto this path.
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  7. #7
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    Quote Originally Posted by olski1 View Post
    Okay thanks for your help so far,

    I am really stuck on incorporating the probability into this question. I cannot seem to find the mathematical reasoning. Although maybe its because its 1:30 in the morning :P
    theodds has given all the reasoning necessary, I think, but as for making a connection between the question and concepts from probability/statistics, it reminds me of absorbing states in Markov chain theory. (Also accepting states for finite state machines.)
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  8. #8
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    Quote Originally Posted by undefined View Post
    theodds has given all the reasoning necessary, I think, but as for making a connection between the question and concepts from probability/statistics, it reminds me of absorbing states in Markov chain theory. (Also accepting states for finite state machines.)
    I figure if you assume sufficient randomness of the lengths of the words you can get at this by looking at partial sums of sequences of integer-valued r.v.'s. Maybe assume word lengths are Poisson distributed and work with that. Something along those lines might work if you iron the details out.

    A twist on the same concept is playing the same game with the digits of your favorite irrational number.
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