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3882

2021-01-07 Bayes’ Theorem shark 1 min read

Consider a test to detect a disease that 0.1 % of the population have. The test is 99 % effective in detecting an infected person. However, the test gives a false positive result in 0.5 % of cases. If a person tests positive for the disease what is the probability that they actually have it?

Solution:

$P(D = 1) = 0.1\% \Rightarrow P(D=0) = 0.999$

$P(T = 1 | D=1) = 99\%$

$P(T = 1 | D=0) = 0.5\%$

$P(D=1 | T=1)= ?$

(1) $\begin{align*}P(T=1) &= P(T=1 | D=1) \times P(D=1) + P(T=1 | D=0) \times P(D=0) \\& = 0.99 \times 0.001 + 0.005 \times 0.999 \\& = 0.005985 \\P(D=1 | T=1) & = \frac{P(D=1) \times P(T=1 | D=1)}{P(T=1)} \\& = \frac{0.001 \times 0.99}{0.005985} \\& = 0.17 \\\end{align*}$

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