Bayes Formula Problem Solution Check

Question

The occurrence of a disease is $\frac{1}{100} = P(D)$

The false negative probability is $\frac{6}{100} = P(- | D)$, and the false positive is $\frac{3}{100} = P(+ | \neg D)$

Compute $P(D | +)$

By bayes formula, $P(+) = P(+ | D) P(D) + P(+ | \neg D) P(\neg D) = \frac{97}{10000} + \frac{297}{10000} = \frac{394}{10000}$

Similarly $P(D | +) = \frac{P(+ | D) P(D)}{P(+)} = \frac{97}{394} = 0.246$

Is this correct?

Answer 1

no

$\mathbb{P}[D|T^+]=\frac{\mathbb{P}[T^+ \cap D]}{\mathbb{P}[T^+ \cap D]+\mathbb{P}[T^+ \cap \overline{D}]}=\frac{(100-6)\times1}{(100-6)\times1+3\times(100-1)}=24.04$

Notes

• I expressed the data in % to avoid lot of decimals, so the result is 24.04%

• All the data expressed in formulas, e.g $(100-1)$, mean that they are calculated and not given by the text