Bayes Theorem to calculate probability

Question

Suppose that you are an art dealer for Van Gogh paintings. Whenever someone brings in a painting that they claim is from Van Gogh, you send in the painting for pigment analysis. Pigment analysis successfully identifies 90% of Van Gogh paintings and falsely identifies 20% of forgeries as being from Van Gogh. In your experience, 99% of the paintings that people bring to you are forgeries.

Use Bayes theorem to calculate the chance that a painting is from Van Gogh if the pigment analysis finds that it is from Van Gogh

Answer 1

Let A = painting is a forgery ,

A' = painting is not a forgery (painting is from Van Gogh)

B=pigment analysis finds it is from VG

B'= pigment analysis says it is not from VG

Given: P(A)=0.99 P(B|A')=0.9 P(B|A)=0.2

Want: P(A'|B) = ?

P(A')=1-P(A) = 1-099=0.01

Now, using Bayes' formula,

P(A'|B) = P(B|A')P(A') / [P(B|A')P(A') + P(B|A)P(A) ]

= (0.9 * 0.01) / (0.9*0.01 + 0.2*0.99)

~ 0.043478