document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=394879>Question 394879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove:  The probability that two diagonals in a convex polygon will intersect inside the polygon is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%5E2+-+3n+%2B+2%29%2F%283%28n%5E2++-3n++-2%29%29 ALIGN=MIDDLE  ALT=\"%28n%5E2+-+3n+%2B+2%29%2F%283%28n%5E2++-3n++-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #280244 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=280244\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Sorry, I had misread the minus as a plus and nearly thought that the probability was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%5E2+-+3n+%2B+2%29%2F%283%28n%5E2++-3n++%2B2%29%29 ALIGN=MIDDLE  ALT=\"%28n%5E2+-+3n+%2B+2%29%2F%283%28n%5E2++-3n++%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies to 1/3 (and doesn't make sense). I have revised my solution here, but didn't have enough time to finish, as there is probably going to be a ton of algebra. However I have got you started, you can finish the algebra from then on.\r<BR>\n" );
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document.write( "I'm quite sure that the best way is to fix one of the n points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D ALIGN=MIDDLE  ALT=\"a%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ..., <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We go case by case on what the second point of diagonal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5B1%5D ALIGN=MIDDLE  ALT=\"D%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is.\r<BR>\n" );
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document.write( "Case 1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5B1%5D ALIGN=MIDDLE  ALT=\"D%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > includes points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D ALIGN=MIDDLE  ALT=\"a%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Then, there is 1 point between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D ALIGN=MIDDLE  ALT=\"a%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and n-3 points from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D ALIGN=MIDDLE  ALT=\"a%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (counting all the numbers 4, 5, ..., n inclusive).\r<BR>\n" );
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document.write( "We can generalize this to say:\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5B1%5D ALIGN=MIDDLE  ALT=\"D%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > comprises of points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bi%5D ALIGN=MIDDLE  ALT=\"a%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-i%29%28i-2%29 ALIGN=MIDDLE  ALT=\"%28n-i%29%28i-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible diagonals. Given this, we can sum them up from i = 3 to i = n-1. If P is the total number of diagonals, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+sum%28%28n-i%29%28i-2%29%2C+i+=+3%2C+n-1%29+=+sum%28-i%5E2+%2B+%28n%2B2%29i+-+2n%2C+i+=+3%2C+n-1%29+=+sum+%28-i%5E2+%2B+%28n%2B2%29i+-+2n%2C+i+=+1%2C+n-1%29+-+%281-n%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > (subtracting the case where n = 1, 2)\r<BR>\n" );
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document.write( "After this, use the sum identities <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%5E2%2C+i+=+1%2C+k%29+=+k%28k%2B1%29%282k%2B1%29%2F6 ALIGN=MIDDLE  ALT=\"sum%28i%5E2%2C+i+=+1%2C+k%29+=+k%28k%2B1%29%282k%2B1%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2C+i+=+1%2C+k%29+=+k%28k%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28i%2C+i+=+1%2C+k%29+=+k%28k%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. After that, it's pretty much algebra bashing time to find an expression for P.\r<BR>\n" );
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document.write( "Suppose Q is the number of sets of two diagonals. Be careful that one of the diagonals of Q must have an endpoint at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (due to our previous assumption).\r<BR>\n" );
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document.write( "The probability that two chosen diagonals intersect is then P/Q, which should turn out to the expected value (however it's a whole lot of algebra and simplifying from here on).<BR>\n" );
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document.write( "The other solution I considered involved mathematical induction. Basically, you show the base case (n=4, trivial), and show that n = k implies n = k+1. However, I had to split this problem into two cases, one case where the diagonals in the (n+1)-gon were contained within the n-gon, and the other case where one diagonal contains the (n+1)th point. However this solution is probably a bit longer than this one.<BR>\n" );
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