document.write( "Question 1170840: A certain disease has an incidence rate of 0.6%. If the false negative rate is 4% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease. \n" ); document.write( "

Algebra.Com's Answer #850866 by ikleyn(52851) $\"\"$ $\"About$

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\n" ); document.write( "A certain disease has an incidence rate of 0.6%. If the false negative rate is 4%
\n" ); document.write( "and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease.
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document.write( "This problem is on conditional probability, and I will put things in order.\r\n" );
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document.write( "Originally, we have a whole population of X people.\r\n" );
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document.write( "0.6% of them, or 0.006*X, have the disease.\r\n" );
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document.write( "The rest, or 0.994*X, do not have the disease (are healthy).\r\n" );
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document.write( "The number of those who actually has the disease AND has test positive is  n = (1-0.04)*(0.006*X) = 0.96*0.006*X.\r\n" );
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document.write( "The number of those who are test positive is the sum  N = 0.96*(0.006*x) + 0.01*(0.994*X).\r\n" );
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document.write( "In this sum, first  addend comes from the set of people having the disease; \r\n" );
document.write( "             second addend comes from the set of people who do not have the disease.\r\n" );
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document.write( "The problem wants you find the conditional probability, which is the ratio n/N.  It is \r\n" );
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document.write( "    P =  = .\r\n" );
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document.write( "Now reduce X in the numerator and in the denominator.  You will get\r\n" );
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document.write( "    P =  =  =  =  = 0.3669  (rounded), or 36.69%.\r\n" );
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document.write( "       So, the effectiveness of such a test is, actually, lower than expected.\r\n" );
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document.write( "The major reason, WHY it is so low, is that the contribution of false positive tests is very big.\r\n" );
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document.write( "When the incidence rate is low, it leads to loss of effectiveness of testing.\r\n" );
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document.write( "When the incidence rate is low, the requirements to the test to be effective are very low false positive rates.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "See also the solution to this problem by another tutor under this link\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1171091.html\r
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