document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195936>Question 1195936</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that 4% of all the patients are infected with the virus (event V),<BR>\n" );
document.write( "P(V) = 0.04.There exists a test for this viral infection. It is 95% reliable for infected patients and<BR>\n" );
document.write( "99% reliable for healthy ones. That is, if a patient has the virus, the test shows that (event S) with<BR>\n" );
document.write( "probability P(S | V) = 0.95, and if the patient does not have the virus, the test shows that with<BR>\n" );
document.write( "probability P(S^c|V^c) = 0.99. If a person is tested positive, what is the probability that he/she is infected? </SPAN>\n" );
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document.write( "It might be easy to get lost with terms like \"false positive\" or \"true negative\".\r<BR>\n" );
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document.write( "Here's a chart to keep them organized.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Positive Test</TD><TD >Negative Test</TD></tr><tr><TD >Virus</TD><TD >True Positive</TD><TD >False Negative</TD></tr><tr><TD >No Virus</TD><TD >False Positive</TD><TD >True Negative</TD></tr>\n" );
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document.write( "In medical testing, \"positive\" is when the test answers \"yes\" to the question \"does the person have the infection/virus/disease/etc?\". <BR>\n" );
document.write( "Unfortunately it might be confusing since many people would consider positive to be a good thing in any other context.\r<BR>\n" );
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document.write( "Consider a city with 100,000 people.\r<BR>\n" );
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document.write( "4% of this population has the virus.<BR>\n" );
document.write( "4% of 100,000 = 0.04*100,000 = 4000 people have the virus.\r<BR>\n" );
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document.write( "The true positive rate is 95%<BR>\n" );
document.write( "This means that if a person has the virus, then the test will result \"positive\" 95% of the time.<BR>\n" );
document.write( "This is what the P(S | V) = 0.95 refers to.\r<BR>\n" );
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document.write( "95% of 4000 = 0.95*4000 = 3800 people with the virus will get a true positive test result.<BR>\n" );
document.write( "The remaining 4000-3800 = 200 people get false negatives.<BR>\n" );
document.write( "Put another way, 5% of 4000 = 0.05*4000 = 200 people get false negatives.\r<BR>\n" );
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document.write( "The 100,000 - 4,000 = 96,000 people who don't have the virus will have 0.99*96,000 = 95,040 people get true negatives.<BR>\n" );
document.write( "The other 96,000 - 95,040 = 960 people will get false positives (i.e. 1% of 96000 = 960)\r<BR>\n" );
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document.write( "Side note:<BR>\n" );
document.write( "The term \"test sensitivity\" refers to the true positive rate.<BR>\n" );
document.write( "The term \"test specificity \" refers to the true negative rate.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Sensitivity_and_specificity\">https://en.wikipedia.org/wiki/Sensitivity_and_specificity</a>\r<BR>\n" );
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document.write( "This current test has a sensitivity of 95% and specificity of 99%<BR>\n" );
document.write( "We can rephrase the test specificity of 99% to be a false positive rate of 1%\r<BR>\n" );
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document.write( "Since we made a chart earlier, let's fill in that table with the numbers previously calculated<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Positive Test</TD><TD >Negative Test</TD><TD >Total</TD></tr><tr><TD >Virus</TD><TD >3800</TD><TD >200</TD><TD >4000</TD></tr><tr><TD >No Virus</TD><TD >960</TD><TD >95,040</TD><TD >96,000</TD></tr><tr><TD >Total</TD><TD >4760</TD><TD >95,240</TD><TD >100,000</TD></tr>\n" );
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document.write( "Take note how the row and column totals are calculated (simply by adding).\r<BR>\n" );
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document.write( "Now we're told that a person tested positive. This is the \"given\".<BR>\n" );
document.write( "We'll focus entirely on the \"positive\" column. Ignore everything else.\r<BR>\n" );
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document.write( "We have 3800 people who test positive and have the virus.<BR>\n" );
document.write( "This is out of 4760 people who test positive overall.\r<BR>\n" );
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document.write( "Therefore, the <font color=red>final answer</font> is approximately <BR>\n" );
document.write( "3800/4760 = <font color=red>0.798319</font><BR>\n" );
document.write( "Or you could stick to the fraction form. Be sure to reduce it as much as possible if your teacher requires that.\r<BR>\n" );
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document.write( "As an alternative approach, you can use <a  rel=nofollow HREF= \"https://www.mathsisfun.com/data/bayes-theorem.html\">Bayes Theorem</a> to solve this question. <BR>\n" );
document.write( "P(A | B) = P(B|A)*P(A)/P(B)<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "A = person has the virus<BR>\n" );
document.write( "B = person tests positive<BR>\n" );
document.write( "The vertical line represents the key word \"given\".<BR>\n" );
document.write( "You'll need the law of total probability to compute P(B)<BR>\n" );
document.write( "Let me know if you have any questions about this pathway.\r<BR>\n" );
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