document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192026>Question 1192026</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 9% of the country's population has symptoms of the disease. After a test in a symptomatic person, it is determined with a probability of 0.78% and for an asymptomatic person, the test gives a positive response with a probability of 0.06%. <BR>\n" );
document.write( "a) what is the probability that the test will be positive for a random person?<BR>\n" );
document.write( "b) what is the probability that a person who tested positive actually has a symptoms of the disease? </SPAN>\n" );
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document.write( "It seems like there are typos when you wrote the 0.78% and 0.06%, as those values seem really really low.\r<BR>\n" );
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document.write( "I'm going to assume you meant to say 0.78 and 0.06 instead, without the percent signs.\r<BR>\n" );
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document.write( "Consider a country with 1,000,000 people.\r<BR>\n" );
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document.write( "9% of them show symptoms, so 0.09*(1,000,000) = 90,000 people show symptoms. <BR>\n" );
document.write( "Of the people who show symptoms, the probability of getting a positive test is 0.78<BR>\n" );
document.write( "This means 0.78*(90,000) = 70,200 symptomatic people get a positive test. This is considered a true positive.\r<BR>\n" );
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document.write( "The remaining 90,000 - 70,200 = 19,800 people get negative tests when it should have been positive. <BR>\n" );
document.write( "This is the false negative scenario.\r<BR>\n" );
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document.write( "Since 90,000 showed symptoms, this means 1,000,000 - 90,000 = 910,000 people are asymptomatic (i.e. they don't show symptoms). <BR>\n" );
document.write( "Whether or not they *actually* have the disease isn't clear because the disease may be present in asymptomatic people. <BR>\n" );
document.write( "Though for the sake of simplicity, I'll assume asymptomatic people are in the clear and don't have the disease.\r<BR>\n" );
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document.write( "Anyway, we have 910,000 asymptomatic people. Of these, the probability of getting a positive test is 0.06<BR>\n" );
document.write( "We expect about 0.06*(910,000) = 54,600 asymptomatic people to get a positive test. This is the false positive scenario, which is basically a false alarm.<BR>\n" );
document.write( "The other 910,000 - 54,600 = 855,400 asymptomatic people get a negative test. This is the true negative scenario.\r<BR>\n" );
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document.write( "Let's summarize:<ul><li>Total population = 1,000,000 people</li><li>90,000 show symptoms and 910,000 do not</li><li>70,200 symptomatic people get positive tests (true positives) while 19,800 symptomatic people get negative tests (false negatives)</li><li>54,600 asymptomatic people get positive tests (false positives) and 855,400 asymptomatic people get negative test results (true negatives)</li></ul>Admittedly, there are a lot of numbers to keep track of even in a summarized list form like that.\r<BR>\n" );
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document.write( "So perhaps a better way is to list it in a two-way table<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Symptoms</TD><TD >No symptoms</TD></tr><tr><TD >Test positive</TD><TD >True positive</TD><TD >False positive</TD></tr><tr><TD >Test negative</TD><TD >False negative</TD><TD >True negative</TD></tr>\n" );
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document.write( "which updates to<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Symptoms</TD><TD >No symptoms</TD><TD >Total</TD></tr><tr><TD >Test positive</TD><TD >70,200</TD><TD >54,600</TD><TD >124,800</TD></tr><tr><TD >Test negative</TD><TD >19,800</TD><TD >855,400</TD><TD >875,200</TD></tr><tr><TD >Total</TD><TD >90,000</TD><TD >910,000</TD><TD >1,000,000</TD></tr>\n" );
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document.write( "After the table is set up, we can then address the questions.\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "We have 124,800 people who test positive out of 1,000,000 total.\r<BR>\n" );
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document.write( "Therefore, <BR>\n" );
document.write( "(124,800)/(1,000,000) = 0.1248<BR>\n" );
document.write( "is the probability of testing positive.\r<BR>\n" );
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document.write( "Answer: <font color=red>0.1248</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Focus on the \"test positive\" row. This is because the prior condition is that we know the person tested positive. <BR>\n" );
document.write( "We're seeking to determine the probability of getting symptoms based on this prior knowledge.\r<BR>\n" );
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document.write( "We have 124,800 people who tested positive.<BR>\n" );
document.write( "Of these people, we have 70,200 present symptoms\r<BR>\n" );
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document.write( "(70,200)/(124,800) = 0.5625<BR>\n" );
document.write( "There's a 56.25% chance of someone presenting symptoms if the test returns positive.\r<BR>\n" );
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document.write( "Answer: <font color=red>0.5625</font><BR>\n" );
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