document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192522>Question 1192522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1 point) It is estimated that approximately 8.37% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 98% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 3% of all adults over 40 without diabetes as having the disease.\r<BR>\n" );
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document.write( "    a) Find the probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called \"false positives\").\r<BR>\n" );
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document.write( "    b) Find the probability that a randomly selected adult of 40 is diagnosed as not having diabetes.\r<BR>\n" );
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document.write( "    c) Find the probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called \"false negatives\").\r<BR>\n" );
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document.write( "(Note: it will be helpful to first draw an appropriate tree diagram modeling the situation) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849045 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52788)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "It is estimated that approximately 8.37% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes <BR>\n" );
document.write( "will correctly diagnose 98% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 3% of all adults over 40 <BR>\n" );
document.write( "without diabetes as having the disease.\r<BR>\n" );
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document.write( "(a) Find the probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes <BR>\n" );
document.write( "(such diagnoses are called \"false positives\").\r<BR>\n" );
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document.write( "(b) Find the probability that a randomly selected adult <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cross%28of%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cross%28of%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <U>over</U> 40 is diagnosed as not having diabetes.\r<BR>\n" );
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document.write( "(c) Find the probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes <BR>\n" );
document.write( "(such diagnoses are called \"false negatives\").<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In the post by @CPhill, all three parts (a), (b) and (c) are solved and answered INCORRECTLY.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I came to bring correct solutions to all parts.\r<BR>\n" );
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document.write( "(a)  In (a), they want you determine the probability of two simultaneous events:\r\n" );
document.write( "     the person does not have diabetes, but is diagnosed incorrectly as having diabetes.\r\n" );
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document.write( "     The <U>ANSWER</U> to (a) is  0.9263*(1-0.98) = 0.018526.\r\n" );
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document.write( "(b)  This probability is the sum of two probabilities of disjoint events:\r\n" );
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document.write( "         - the person does not have diabetes and diagnosed correctly as not having diabetes\r\n" );
document.write( "               P1 = 0.9263*0.98;\r\n" );
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document.write( "         - the person has diabetes, but diagnosed incorrectly as not having diabetes\r\n" );
document.write( "               P2 = 0.0837*0.(1-0.03).\r\n" );
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document.write( "      Therefore, the probability in (b) is  P = P1 + P2 = 0.9263*0.98 + 0.0837*(1-0.03) = 0.988963.  <U>ANSWER</U> to (b)\r\n" );
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document.write( "(c)  In (c), the conditional probability is \r\n" );
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document.write( "          P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0837%2F%280.9263%2A0.98+%2B+0.0837%2A%281-0.03%29%29 ALIGN=MIDDLE  ALT=\"0.0837%2F%280.9263%2A0.98+%2B+0.0837%2A%281-0.03%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.0837%2F0.988963 ALIGN=MIDDLE  ALT=\"0.0837%2F0.988963\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0.084634.  <U>ANSWER</U> to (c)\r\n" );
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document.write( "     In this formula, the numerator is the probability of the event that the person has diabetes. \r\n" );
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document.write( "     The denominator is the probability that he/she is diagnosed as not having diabetes.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "The goal of this problem is to teach a student to think logically in this specific area.<BR>\n" );
document.write( "Having and using common sense is enough to make every step.\r<BR>\n" );
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document.write( "Knowing the formal theorems is not required.<BR>\n" );
document.write( "Referring to these formal theorems is not required, too.<BR>\n" );
document.write( "It only distracts attention from the solution.\r<BR>\n" );
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document.write( "An example on how incorrectly the problem was solved in the post by @CPhill shows<BR>\n" );
document.write( "that the references to formal theorems do not save from making monstrous errors.<BR>\n" );
document.write( "Do not save from making monstrous errors at every step.\r<BR>\n" );
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document.write( "Common sense should work and really works much better in such simple problems.<BR>\n" );
document.write( "And this is exactly my major goal in this post to help students to develop their common sense.\r<BR>\n" );
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document.write( "And it should be the major goal of every teaching on how to solve such problems.\r<BR>\n" );
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document.write( "And not at all to weave a web of references to theorems in order to demonstrate your knowledgeability.\r<BR>\n" );
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document.write( "About making errors in such problems during their solution.\r<BR>\n" );
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document.write( "I think it is normal to make 5 errors at first attempt; <BR>\n" );
document.write( "3 errors at the second trial, 2 errors at the third trial, <BR>\n" );
document.write( "1 error at the fourth trial and 0 (zero) errors at fifth trial.\r<BR>\n" );
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document.write( "After that, I would recommend re-solve the problem three more times <BR>\n" );
document.write( "to make sure that all errors are just fixed and there are no new errors.\r<BR>\n" );
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