document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=372000>Question 372000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If there are 60 questions on an exam, the succes rate is 1/4. You know 24 out of 60 questions, what is the probability you will suceed on the exam if the passing grade is 50%. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #264992 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=264992\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> successes in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on any given trial is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n(k,p)\ =\ \left(n\cr k\right\)p^k\left(1\,-\,p\right)^{n\,-\,k}\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr k\right\)\"> is the number of combinations of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> at a time and is calculated by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{n!}{k!(n\,-\,k)!}\">\r<BR>\n" );
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document.write( "Given that you know 24 out of the 60, you need the probability of guessing at least 6 out of 36 with a per question probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{4}\">.\r<BR>\n" );
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document.write( "You can go about this one of two ways.  The probability of at least 6 correct out of 36 is the probability of exactly 6 plus exactly 7 plus exactly 8 ... plus exactly 35, plus exactly 36.  Symbolically:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{36}\left(\geq6,\frac{1}{4}\right)\ =\ \sum_{i=6}^{36}\left(36\cr\ i\right\)\left(\frac{1}{4}\right)^i\left(\frac{3}{4}\right)^{36\,-\,i}\"> \r<BR>\n" );
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document.write( "Which, trust me, is a very long, tedious, and therefore highly error prone calculation in that the sum consists of 31 terms, several of which have coefficients in the billions.\r<BR>\n" );
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document.write( "However, there is an easier way.  Note that the probability of getting 5 or less correct PLUS the probability of getting 6 or more correct is just 1, or certainty -- if you guess at 36 questions it is an absolute certainty that you will get between 0 and 36 of them correct.  Hence the desired probability is 1 minus the probability of getting 5 or less correct:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{36}\left(\geq6,\frac{1}{4}\right)\ =\ 1\ -\ P_{36}\left(\leq5,\frac{1}{4}\right)\ =\ 1\ -\ \sum_{i=0}^{5}\left(36\cr\ i\right\)\left(\frac{1}{4}\right)^i\left(\frac{3}{4}\right)^{36\,-\,i}\">\r<BR>\n" );
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document.write( "which sum only has 7 terms (counting the 1) and the largest coefficient is 58905.  Get out your calculator and get cracking.\r<BR>\n" );
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document.write( "You should get an answer that is between 91 and 92 percent.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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