document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=974803>Question 974803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you apply the Bayes' Theorem?  Thank you. \r<BR>\n" );
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document.write( "Use Bayes Theorem for the following <BR>\n" );
document.write( "In the current second year class of  a top ranked US university, all students come from three states in the US . Kansas, Illinois, and  Ohio supply respectively 40%, 50%, and 10%  of the students. The failure rate of students is 4%, 2%, and 6% respectively.\r<BR>\n" );
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document.write( "What is the probability that a student chosen at random will fail?<BR>\n" );
document.write( "Given that a student fails , what is the probability that he or she came form Kansas?<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 #596618 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=596618\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A_i\"> denote the event that a randomly chosen student comes from state <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i\ \in\ \{1, 2, 3\}\"> and 1 represents Kansas, 2 represents Illinois, and 3 represents Ohio.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B\"> represent the event that a randomly chosen student will fail. We are given the following infomation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(A_1)\ =\ 0.4\ \ \ \"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(A_2)\ =\ 0.5 \ \ \ \"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(A_3)\ =\ 0.1\">\r<BR>\n" );
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document.write( "If a student is from Kansas, then the probability that student will fail is 0.04. That is to say:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(B\ |\ A_1)\ =\ 0.04\">.  By the same analysis, we have, overall:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(B\ |\ A_1)\ =\ 0.04\ \ \ \"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B\ |\ A_2)\ =\ 0.02 \ \ \ \"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B\ |\ A_3)\ =\ 0.06\">\r<BR>\n" );
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document.write( "To answer the first question, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B\ =\ ?\"> we use the following:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(B)\ =\ \sum_{i=1}^3\, P(B\ |\ A_i)\,\cdot\,P(A_i)\ =\ P(B\ |\ A_1)\,\cdot\,P(A_1)\ +\ P(B\ |\ A_2)\,\cdot\,P(A_2)\ +\ P(B\ |\ A_3)\,\cdot\,P(A_3)\">\r<BR>\n" );
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document.write( "The arithmetic is left as an exercise for the student.\r<BR>\n" );
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document.write( "Second question: Given that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large B\"> has occurred, we then want to calculate the conditional probability of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A_1\">.  By Bayes' Theorem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(A_1\ |\ B)\ =\ \frac{P(B\ |\ A_1)\,\cdot\,P(A_1)}{P(B)}\">\r<BR>\n" );
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document.write( "The rest is arithmetic.\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\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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