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How to Use Random Cut Theorem and Simple Probability
In Euclid's "Elements" are several versions of the Random Cut Theorem; here you will see one of the simpler ones (Book II Proposition 4) to make a point about simple probability you can use.
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EditSteps
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1Learn or recall from "Elements", Book II, Proposition 4, that "If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments."- Grab/copy the yellow and orange diagram from this page onto the Clipboard.
- TYpe Command and c for copy to make a copy of the diagram.
- At the Desktop, click on the XL icon on the Dock to start Excel.
- Open a New Workbook in Excel.
- Holding down the Shift Key, do Edit Paste Picture into a new worksheet.
- Save the file under an appropriate filename into a logical folder, and make notes from the steps below, below the diagram.
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2Learn or recall that, per the above diagram, if segment AC = x and segment CB = y, that (x+y)^2 also equals x^2 + y^2 + 2xy. Stated another way, if x = y+z, then x^2 = (y+z)^2 = y^2 + 2yz + z^2. In the diagram then, square HF = y^2, square CK = z^2, rectangle AG = yz and rectangle GE = yz as well. -
3Call the Area of the Whole square AE equal to 1 or 100% Probability. -
4Guesstimate from the random cut in the diagram that it occurred at a value of .70 of length AB. -
5Calculate from your guesstimate what a dart's chance's of landing in each square are, or where the next robbery will be, i.e. where the next random event will be.- There is a .3 * .7 = 21% chance the next event will be in one of the two rectangles, and so a 42% chance it will be in either rectangle (or in 2yz).
- There is a .3 *.3 = 9% chance the next event will be in square CK (or in z^2).
- There is a .7 * .7 = 49% chance the next event will be in square HF (or in y^2).
- (.42 + .09 + .49 = 1.00 = 100% chance there will be a next event (assumably). This is an important assumption and not always the case. Or perhaps it's a presumption. It's in the sump somewhere, speaking etymologically ...
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EditTips
- Note that if there's an equation like (2.47 + 6.03x)^2, the numbers can be summed and pro-rated to determine the likelihood of a result being a cause of one of the quantities. Thus, 2.47 + 6.03 = 8.5 and 2.47/8.5 = .2906 and 6.03/8.5 must be .7094; these we can combine as above into our areas of y^2, 2yz and z^2. A result following from this formula will fall somewhere into one of those 3 classes, according to the probability of each, but even more so with the x attached as a factor to one member, as that will skew the results further from what they'd otherwise be (unless x=1, or both variables equate to 0).
EditWarnings
- Do not forget to be reasonable. If the event was a "cut", like an axe cut, another one may soon fall quite nearby. Look for a CAUSE and EFFECT relationship before going hogwild with statistics. If the situation was that y was bound to z, or made fixed in an AND in some way, then maybe an extension at either end or again a branching at the cutpoint is probable. Learn to think outside the box in other words. And if y+z is determined to be part of a series, say z = y/(y-1) so that y+z = y*z, then look for the series to continue.
EditRelated wikiHows
EditSources and Citations
- http://archive.org/details/thefirstsixbooks21076gut Creative Commons license
Article Info
Categories: Geometry | Probability and Statistics
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