document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197221>Question 1197221</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A truth serum given to a suspect is known to be 90 % reliable when the person is guilty and 99 % reliable when the person is innocent. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent? </SPAN>\n" );
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document.write( "Think of an infection rates problem such as this one<BR>\n" );
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document.write( "I'll be following the same type of outline but with a few slight modifications.\r<BR>\n" );
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document.write( "positive test = serum indicates the person is guilty<BR>\n" );
document.write( "negative test = serum indicates the person is innocent\r<BR>\n" );
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document.write( "Consider a town of 10,000 people.\r<BR>\n" );
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document.write( "5% of this town has committed a crime<BR>\n" );
document.write( "5% of 10,000 = 0.05*10,000 = 500 people committed a crime\r<BR>\n" );
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document.write( "Of the 500 people who committed a crime, the truth serum will return a guilty verdict correctly 90% of the time<BR>\n" );
document.write( "90% of 500 = 0.90*500 = 450 people test positive<BR>\n" );
document.write( "We consider this the true positives.\r<BR>\n" );
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document.write( "The remaining 500-450 = 50 people who committed a crime get \"not guilty\" verdicts incorrectly. These are false negatives.\r<BR>\n" );
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document.write( "Since this town has 10,000 people and 500 committed a crime, that leaves 10,000-500 = 9500 people who didn't commit a crime.<BR>\n" );
document.write( "The test is 99% reliable when the person is innocent.<BR>\n" );
document.write( "Meaning 0.99*9500 = 9405 truly innocent people will get a correct \"not guilty\" verdict. These are the true negatives.\r<BR>\n" );
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document.write( "The remaining 9500-9405 = 95 people get a false positive. The serum says guilty, but the person is actually innocent.\r<BR>\n" );
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document.write( "Here's a chart to help remember the various terms discussed.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Positive Test</TD><TD >Negative Test</TD></tr><tr><TD >Crime</TD><TD >True positive</TD><TD >False negative</TD></tr><tr><TD >No Crime</TD><TD >False positive</TD><TD >True negative</TD></tr>\n" );
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document.write( "And here's the chart of values for this specific problem. Each value was calculated in previous paragraphs.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Positive Test</TD><TD >Negative Test</TD><TD >Total</TD></tr><tr><TD >Crime</TD><TD >450</TD><TD >50</TD><TD >500</TD></tr><tr><TD >No Crime</TD><TD >95</TD><TD >9405</TD><TD >9500</TD></tr><tr><TD >Total</TD><TD >545</TD><TD >9455</TD><TD >10,000</TD></tr>\n" );
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document.write( "-------------------------------------------------\r<BR>\n" );
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document.write( "With all that set up in mind, we're then asked this question:<BR>\n" );
document.write( "\"If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?\"\r<BR>\n" );
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document.write( "In other words, we're given the serum states \"guilty\".<BR>\n" );
document.write( "So we can narrow our focus on just the \"positive test\" cases only.<BR>\n" );
document.write( "There are 450 correct guilty verdicts (true positive) and 95 incorrect guilty verdicts (false positive)<BR>\n" );
document.write( "450+95 = 545 guilty verdicts in all\r<BR>\n" );
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document.write( "The probability of the person being innocent is 95/545 = 0.1743 = 17.43% approximately\r<BR>\n" );
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document.write( "Round the decimal and percentage values however your teacher instructs.\r<BR>\n" );
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document.write( "Side note: The fraction 95/545 fully reduces to 19/109<BR>\n" );
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document.write( " </SPAN>\n" );
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