document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=261826>Question 261826</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I understand basic probability but am very confused when it comes to the below question.  I don't even have a clue where to start to solve this question.\r<BR>\n" );
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document.write( "	A multiple choice test has 20 questions with each having 4 possible answers with one correct. Assume a student randomly guesses the  answer to every question. \r<BR>\n" );
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document.write( "a. What is the probability of getting exactly 11 correct answers?\r<BR>\n" );
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document.write( "20x4=80 1 right answer from 1 question=1/4\r<BR>\n" );
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document.write( "b. What is the probability of getting less than 9 correct answers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #192919 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=192919\">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( "You need the probability of <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 success on any given trial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> and the probability of failure on any given trial is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large q\">\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^kq^{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 ways to select <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> things from <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things which is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{n!}{k!\left(n\,-\,k\right)!}\">\r<BR>\n" );
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document.write( "Your probability of success for a 4 choice test item is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{4}\">.  Consequently your probability of failure for a 4 choice test item is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\ -\ \frac{1}{4}\ =\ \frac{3}{4}\">.\r<BR>\n" );
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document.write( "Your <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> is 20 because there are 20 questions and your <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> is 11 because you want the probability of exactly 11 correct.  So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{20}\left(11\right)\ =\ \left(20\cr 11\right)\left(\frac{1}{4}\right)^{11}\left(\frac{3}{4}\right)^{9}\ =\ \frac{20!}{11!9!}\left(\frac{1}{4}\right)^{11}\left(\frac{3}{4}\right)^{9}\">\r<BR>\n" );
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document.write( "Looks like it is time to get out the calculator.\r<BR>\n" );
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document.write( "For part b.  You need the probability of all possibilities of getting less than 9 correct.  So, that is the probability of getting 0 correct plus the probability of getting 1 correct plus...and so on...plus the probability of getting 8 correct.  Figure out each of those probabilities the same way as we did in part a and then sum them.\r<BR>\n" );
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document.write( "The actual formula for part b is:\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)_{cum}\ =\ \sum_{i=0}^k{\left(n\cr i\right)p^iq^{n-i}}\">\r<BR>\n" );
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document.write( "If you just want the answers quickly and you have access to MS Excel, then open a blank spreadsheet and, for part a, type the following into any blank cell:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">=BINOMDIST(11,20,0.25,FALSE)\r<BR>\n" );
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document.write( "And for part b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">=BINOMDIST(8,20,0.25,TRUE)\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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