document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=575425>Question 575425</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am an instructor at a high school and just gave a fifty item multiple choice exam (a,b,c,d).  Five students who sit together at a table got the same 8 of the 50 questions wrong.  What is the probability of that happening?  I want to be able to support the fact that I believe they cheated.  Thanks!\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #369517 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=369517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll assume that those five students (A,B,...E) got the other 42 correct.\r<BR>\n" );
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document.write( "Suppose we fix student A's set of responses. We want to find the probability that student B got the exact same set of responses, by chance. If the probability of obtaining a question is 1/5, it is pretty simple:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B = A) = (\frac{1}{5})^{42} (\frac{4}{5})^8\">, we don't multiply by 50C8 because it's not exactly a binomial distribution; order of the problems is important. This probability is roughly 7.37*10^(-31), pretty small. Since we also have students C,D,E, raise this number to the fourth power; you'll get a pretty minuscule number.\r<BR>\n" );
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document.write( "However, the probability of a correct answer is not 20%. Assume the probability of a correct answer is p, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B = A) = p^{42}(1-p)^8\">\r<BR>\n" );
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document.write( "If p = .99, then P(B=A) = 6.56*10^(-17). You can use calculus or inequalities to optimize P(B = A) but it will still be small. Note that you have to raise this probability to the fourth power to account for four students who are supposedly independent of each other. They're probably cheating.\r<BR>\n" );
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document.write( "There are definitely exceptions though, e.g. the eight questions were extremely difficult while the other 42 were extremely easy. Then I wouldn't be all that suspicious. </SPAN>\n" );
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