document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316540>Question 316540</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A student takes a 20-question, true/false exam and guesses on each question.  fkind the probability of passing if the lowest passing grade is 15 correct out of 20.  How do I compute using the binomial distribution formula: <BR>\n" );
document.write( "P(X) = n!         *p<BR>\n" );
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document.write( "     (n-x)!X! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226459 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=226459\">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 basic formula for the probability of exactly <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> successes out of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> trials where the probability of a successful outcome on a single trial is fixed at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> 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)\ =\ \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:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(n\cr k\right\)\ =\ \frac{n!}{k!(n\ -\ k)!}\">\r<BR>\n" );
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document.write( "But you need the probability of 15 right answers plus 16 right answers plus ... plus 19 right answers plus 20 right answers.\r<BR>\n" );
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document.write( "So you need:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{20}(\geq15)\ =\ \sum_{i=15}^{20}\left(20\cr \ i\right\)0.5^i\left(1\,-\,0.5\right)^{n\,-\,i}\ =\ \sum_{i=15}^{20}\left(20\cr \ i\right\)0.5^i\left(0.5\right)^{n\,-\,i}\  \">\r<BR>\n" );
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document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0.5^i\left(0.5\right)^{n\,-\,i}\ =\ 0.5^n\">, we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{20}(\geq15)\ =\ \left(\left(20\cr 15\right\)+\ \left(20\cr 16\right\)\ +\left(20\cr 17\right\)\ +\left(20\cr 18\right\)\ +\left(20\cr 19\right\)\ +\left(20\cr 20\right\)\right)0.5^{20}\">\r<BR>\n" );
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document.write( "Get busy.  You have some serious calculator punching to do.  Or you can use Excel.  The formula:  =COMBIN(N,K) where N and K are the values you want gives you the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(n\cr k\right\)\">\r<BR>\n" );
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document.write( "I ran this out and got a little over 2% probability.\r<BR>\n" );
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document.write( "SUPER-DOOPER DOUBLE PLUS EXTRA CREDIT.\r<BR>\n" );
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document.write( "10 question multiple-choice exam.  All questions have four possible responses.  Passing grade is 6 or better correct.  Probability you can pass by just guessing?\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" );
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