document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=875108>Question 875108</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 10% of the units in a batch have a common defect. Experience show that 87% of the defective units posses a certain characteristic, while only 3% of the units which do not have this defect posses that characteristic. A unit is examined and found to have the characteristic. What's the conditional probability that the unit has the defect? Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #527970 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=527970\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Events:\r<BR>\n" );
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document.write( "C: item possesses a certain characteristic<BR>\n" );
document.write( "C': item does not posses a certain characteristic<BR>\n" );
document.write( "D: item has a defect<BR>\n" );
document.write( "D': item does not have a defect\r<BR>\n" );
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document.write( "Probabilities:\r<BR>\n" );
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document.write( "P(C): probability that you choose an item that possesses a certain characteristic<BR>\n" );
document.write( "P(C'): probability that you choose an item that does not possess a certain characteristic<BR>\n" );
document.write( "P(D): probability that you choose an item that is defective<BR>\n" );
document.write( "P(D'): probability that you choose an item that is not defective\r<BR>\n" );
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document.write( "P(C|D): probability that you choose an item that possesses a certain characteristic given the item has a defect<BR>\n" );
document.write( "P(C'|D): probability that you choose an item that does not possess a certain characteristic given the item has a defect<BR>\n" );
document.write( "P(C|D'): probability that you choose an item that possesses a certain characteristic given the item does not have a defect<BR>\n" );
document.write( "P(C'|D'): probability that you choose an item that does not possess a certain characteristic given the item does not have a defect\r<BR>\n" );
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document.write( "P(D|C): probability that you choose an item that is defective given the item has that certain characteristic<BR>\n" );
document.write( "P(D'|C): probability that you choose an item that is not defective given the item has that certain characteristic<BR>\n" );
document.write( "P(D|C'): probability that you choose an item that is defective given the item does not have a certain characteristic<BR>\n" );
document.write( "P(D'|C'): probability that you choose an item that is not defective given the item does not have a certain characteristic\r<BR>\n" );
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document.write( "Given Probabilities\r<BR>\n" );
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document.write( "\"10% of the units in a batch have a common defect\"<BR>\n" );
document.write( "P(D) = 0.10<BR>\n" );
document.write( "P(D') = 0.90 (1 - 0.10 = 0.90)\r<BR>\n" );
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document.write( "\"Experience show that 87% of the defective units posses a certain characteristic\"<BR>\n" );
document.write( "P(C|D) = 0.87 (given a defective unit, 87% of them will have this characteristic)\r<BR>\n" );
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document.write( "\"only 3% of the units which do not have this defect posses that characteristic\"<BR>\n" );
document.write( "P(C|D') = 0.03 (given a non-defective unit, 3% of them will have this characteristic)\r<BR>\n" );
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document.write( "\"A unit is examined and found to have the characteristic. What's the conditional probability that the unit has the defect?\" so we want to find P(D|C). This is the probability of event D happening given event C has happened.\r<BR>\n" );
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document.write( "We want to find P(D|C)\r<BR>\n" );
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document.write( "Use <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Bayes'_theorem\">Baye's Theorem</a> to get\r<BR>\n" );
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document.write( "P(D|C) = (P(C|D)*P(D))/(P(C))\r<BR>\n" );
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document.write( "Then use the <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Law_of_total_probability\">Law of Total Probability</a> to break up P(C) into P(C|D)*P(D)+P(C|D')*P(D')\r<BR>\n" );
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document.write( "We will now have this equation\r<BR>\n" );
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document.write( "P(D|C) = (P(C|D)*P(D))/(P(C|D)*P(D)+P(C|D')*P(D'))\r<BR>\n" );
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document.write( "Plug in the values: P(D) = 0.10, P(D') = 0.90, P(C|D) = 0.87, P(C|D') = 0.03 and evaluate\r<BR>\n" );
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document.write( "P(D|C) = (P(C|D)*P(D))/(P(C|D)*P(D)+P(C|D')*P(D'))<BR>\n" );
document.write( "P(D|C) = (0.87*0.10)/(0.87*0.10+0.03*0.90)<BR>\n" );
document.write( "P(D|C) = <font color=\"red\">0.76315789473684</font><BR>\n" );
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