document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166513>Question 1166513</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A disease test has a false positive rate of 0.5% and a false negative rate of 12%.  Compute the probability that someone who tests negative does not have the virus for the following case:<BR>\n" );
document.write( "Using the test on a population, where the probability of someone having the disease is 6/100000. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852747 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2030)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852747\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classic application of **Bayes' Theorem** to find the posterior probability.\r<BR>\n" );
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document.write( "The probability that someone who tests negative does not have the virus is approximately **99.9999%** (or **0.999999**).\r<BR>\n" );
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document.write( "## 🔬 Probability Calculation\r<BR>\n" );
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document.write( "Let $D$ be the event that a person **Has the Disease**.<BR>\n" );
document.write( "Let $D'$ be the event that a person **Does Not Have the Disease**.<BR>\n" );
document.write( "Let $T'$ be the event that the test result is **Negative**.\r<BR>\n" );
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document.write( "### 1. Identify the Given Probabilities\r<BR>\n" );
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document.write( "* **Prevalence ($P(D)$):** The probability of having the disease in the population.<BR>\n" );
document.write( "    $$P(D) = \frac{6}{100000} = 0.00006$$<BR>\n" );
document.write( "* **Probability of Not Having the Disease ($P(D')$):**<BR>\n" );
document.write( "    $$P(D') = 1 - P(D) = 1 - 0.00006 = 0.99994$$\r<BR>\n" );
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document.write( "* **False Positive Rate ($FP$):** The probability of testing positive when the person does *not* have the disease.<BR>\n" );
document.write( "    $$P(T|D') = 0.5\% = 0.005$$<BR>\n" );
document.write( "* **False Negative Rate ($FN$):** The probability of testing negative when the person *does* have the disease.<BR>\n" );
document.write( "    $$P(T'|D) = 12\% = 0.12$$\r<BR>\n" );
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document.write( "### 2. Required Conditional Probabilities\r<BR>\n" );
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document.write( "We need the probability of correctly testing negative ($P(T'|D')$):<BR>\n" );
document.write( "* **True Negative Rate ($TN$):** The probability of testing negative when the person does *not* have the disease.<BR>\n" );
document.write( "    $$P(T'|D') = 1 - P(T|D') = 1 - 0.005 = 0.995$$\r<BR>\n" );
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document.write( "### 3. Goal\r<BR>\n" );
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document.write( "We want to compute the probability that someone who tests negative does not have the disease, $P(D'|T')$.\r<BR>\n" );
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document.write( "$$P(D'|T') = \frac{P(T'|D') \cdot P(D')}{P(T')}$$\r<BR>\n" );
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document.write( "### 4. Calculate the Total Probability of Testing Negative ($P(T')$)\r<BR>\n" );
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document.write( "The Law of Total Probability states that a negative result can occur in two mutually exclusive ways: a true negative ($T'$ and $D'$) or a false negative ($T'$ and $D$).\r<BR>\n" );
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document.write( "$$P(T') = P(T'|D') \cdot P(D') + P(T'|D) \cdot P(D)$$\r<BR>\n" );
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document.write( "Substitute the values:<BR>\n" );
document.write( "$$P(T') = (0.995 \cdot 0.99994) + (0.12 \cdot 0.00006)$$<BR>\n" );
document.write( "$$P(T') = 0.9949403 + 0.0000072$$<BR>\n" );
document.write( "$$P(T') = 0.9949475$$\r<BR>\n" );
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document.write( "### 5. Apply Bayes' Theorem\r<BR>\n" );
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document.write( "Now, substitute $P(T')$ back into the formula for $P(D'|T')$:\r<BR>\n" );
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document.write( "$$P(D'|T') = \frac{P(T'|D') \cdot P(D')}{P(T')}$$<BR>\n" );
document.write( "$$P(D'|T') = \frac{0.995 \cdot 0.99994}{0.9949475}$$<BR>\n" );
document.write( "$$P(D'|T') = \frac{0.9949403}{0.9949475}$$<BR>\n" );
document.write( "$$P(D'|T') \approx \mathbf{0.9999927}$$\r<BR>\n" );
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document.write( "Rounding to six decimal places, the probability is **0.999993**. </SPAN>\n" );
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