False positive, false negative

Suppose the likelihood of someone being allergic to cats is 1/1000 of the general population. A test to determine if you're allergic to cats has a false positive rate of 5%; that is, 5% of the time, the test will indicate incorrectly that you are allergic to cats. Also, the test is said to be 98% accurate for those who are indeed allergic.

Considering a random individual being tested for the unfortunate cat allergy, what is the probability that if the person tested positive, they are indeed allergic? (In other words, what is the probability the person is actually allergic, given they tested positive? ) Round the solution to the nearest percent.

Correct answer:

p =  2 %

Step-by-step explanation:

$$\begin{gathered} p_0=\frac{1}{1000}=0.001 \\ f=5\%=\frac{5}{100}=\frac{1}{20}=0.05 \\ t=98\%=\frac{98}{100}=\frac{49}{50}=0.98 \\ PP=t\cdot p_0=0.98\cdot 0.001=\frac{49}{50000}\doteq 0.001 \\ PN=1-PP=1-0.001\doteq 0.999 \\ NP=f\cdot (1-p_0)=0.05\cdot (1-0.001)\doteq 0.05 \\ NN=1-NP=1-0.05\doteq 0.9501 \\ P=\frac{PP}{PP+NP}=\frac{0.001}{0.001+0.05}=\frac{98}{5093}\doteq 0.0192 \\ p=100\cdot P=100\cdot 0.0192=2\% \end{gathered}$$

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