Conditional probability with infected bats Assume 20% of a bat population is infected by a certain disease. Bats are classified as infected or non-infected by a particular test which is not completely reliable. Assume the probability to classify a bat as non-infected if it is infected is 0.10. Assume the probability to classify a bat as infected if it is non-infected is 0.05. Determine the probability that a bat is infected if it is classified as non-infected. What I've done: I = infected, C = classified as infected $$ P(I|\overline C) =\frac {P(I \bigcap\overline C)}{P(\overline C)} = \frac{.2 * .1}{.1} = .2 $$ I know the correct answer is 0.0256. What am I doing wrong?

Bayes' theorem gives $$ P(I|\overline{C})={P(\overline{C}|I)\times P(I)\over P(\overline{C})} $$ $$ P(\overline{C}|I)\times P(I)=0.1\times0.2 $$

$$ P(\overline{C}) = P(\overline{C}|I)\times P(I) + P(\overline{C}|\overline{I})\times P(\overline{I}) $$ $$ =0.1\times0.2+ 0.95\times0.8 $$

$$ P(I|\overline{C}) = 0.1\times0.2/(0.1\times0.2+0.95\times0.8)=0.0256. $$