A laboratory blood test is 98% effective in detecting a certain disease if the person has the disease (sensitivity). However, the test also yields a "false positive" result for 0.5% of the healthy persons tested. (That is, if a healthy person is tested, then, with probability 0.005, the test result will show positive.) Assume that 2% of the population actually has this disease (prevalence). What is the probability a person has the disease given that the test result is positive?

#### Solution

Let $B_1$ be the event that a person has the disease, let $B_2$ be the event that a person does not have the disease and let A be the event that the test result is positive.

Given that 2% of the population actually has this disease (prevalence), that is `$P(B_1) = 0.02$`

, `$P(B_2) = 1 - P(B_1) = 0.98$`

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Also a laboratory blood test is 98% effective in detecting a certain disease if the person has the disease (sensitivity). That is `$P(A|B_1) = 0.98$`

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And the test also yields a "false positive" result for 0.5% of the healthy persons tested. That is `$P(A|B_2) = 0.005$`

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The probability a person has the disease given that the test result is positive is

` $$ \begin{aligned} P(B_1|A) &= \frac{P(A|B_1)P(B_1)}{P(A|B_1)P(B_1)+P(A|B_2)P(B_2)}\\ &=\frac{(0.98)(0.02)}{(0.98)(0.02)+(0.005)(0.98)}\\ &=\frac{0.0196}{(0.0196)(0.0049)}\\ &=0.8 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators