document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=278989>Question 278989</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the probability of getting at least one correct answer on a 20-question test with 4 possible answers per question? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #202937 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=202937\">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 hard way to do this is to compute the probability of getting exactly 1 right, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_{20}\left(1\right)\">, then the probability of getting exactly 2 right, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_{20}\left(2\right)\">, then 3, then 4, and so on up to 20, and finally adding all of these probabilities together.  All of that being a big pain in the sit-down because each of the calculations involves the following formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_n\left(k\right)\ =\ \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 n\"> is the number of trials, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> is the number of successes, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> is the probability of success on an individual trial,  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (1\,-\,p)\"> is the probability of failure on an individual trial, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(n\cr k\right)\"> is the number of ways to choose <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> things from a collection of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things and is equal to <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( "All of which means that you have 19 complex calculations plus a 19-term addition problem.  Ah, but there truly is a better way.\r<BR>\n" );
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document.write( "The opposite case of getting at least one right is getting none right.  So the probability of getting at least one right plus the probability of getting none right is equal to 1.  So, just calculate the probability of getting none of them right and subtract that from 1.\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{20}\left(\geq1\right)\ =\ 1\ -\ P_{20}\left(0\right)\ =\ 1\ -\ \left(\frac{20!}{0!(20\,-\,0)!}\right)\left(\frac{1}{4}\right)^0\left(1\,-\,\frac{1}{4}\right)^{20-0}\ =\ 1\ -\ \left(\frac{3}{4}\right)^{20}\">\r<BR>\n" );
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document.write( "Hint: Remember <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0!\ =\ 1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^0\ =\ 1\">  Also:  0.75 \"x^y\" 20 = on your Windows built-in calculator in Scientific mode will give you <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{3}{4}\right)^{20}\">\r<BR>\n" );
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document.write( "The rest of the arithmetic is yours to do.\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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