Suppose that you are considering purchasing a painting allegedly painted by Pica

Question

Suppose that you are considering purchasing a painting allegedly painted by Picasso. Forgeries are so common that only one out go every four pictures allegedly painted by Picasso is, indeed, a genuine Picasso. To get more information, you take the painting to an art museum to have it examined. The curator claims that in 90% of the cases in which she is asked to examine a painting that is genuine, she correctly identifies it as genuine. But in 15% of the cases in which she is given a forgery to examine, she identifies it as genuine. What is the probability that the painting is a forgery given that the curator says that it is a forgery?

Explanation / Answer

P(painted by Picasso)=1/4=0.25

P(not painted by Picasso)=1-0.25=0.75

P(identified genuine|painted by Picasso)=0.9

P(identified genuine|not painted by Picasso)=0.15 thus, P(not identified as genuine |not painted by Picasso)=1-0.15=0.85

Thus, P(identified genuine)=P(identified genuine|painted by Picasso)*P(painted by Picasso) + P(identified genuine|not painted by Picasso)*P(not painted by Picasso)= 0.9*0.25+0.15*0.75=0.3375

Thus, P(not identified genuine)=1-P(identifiedgenuine)=1-0.3375 =0.6625

P(not painted by Picasso|not identified as genuine)=P(not identified as genuine |not painted by Picasso)*P(not painted by Picasso)/P(not identified genuine)

=0.85*0.75/0.6625

=0.9623

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