Question

Imagine that you are a physician and you have just received the results back for a patient of yours who has just tested positive for the “heartbreak of psoriasis”. The test used will correctly label a person who is suffering from the “heartbreak of psoriasis” as a sufferer 90% of the time and will correctly label a person who is not suffering from the “heartbreak of psoriasis” as not being a sufferer 60% of the time. If the base-rate of suffering from the “heartbreak of psoriasis” is 5%, explain to your patient how likely she is actually suffering from the “heartbreak of psoriasis” on the basis of this positive result.

I got 7.32% using Bayes Theorem. Is this right?

Answer 1

P[ suffering from the “heartbreak of psoriasis” ] = 5% = 0.05

P[ not suffering from the “heartbreak of psoriasis” ] = 1 - 0.05 = 0.95

P[ test used will correctly label a person who is suffering from the “heartbreak of psoriasis” as a sufferer ] = 90% = 0.9

P[ will correctly label a person who is not suffering from the “heartbreak of psoriasis” as not being a sufferer ] = 60% = 0.6

P[ will incorrectly label a person who is not suffering from the “heartbreak of psoriasis” as being a sufferer ] = 1 - 0.6 = 0.4

P[ she is actually suffering from the “heartbreak of psoriasis” on the basis of this positive result ] = P[ she is actually suffering from the “heartbreak of psoriasis” | positive result ]

P[ she is actually suffering from the “heartbreak of psoriasis” | positive result ] = P[ she is actually suffering from the “heartbreak of psoriasis” and positive result ]/P[ positive result ]

P[ positive result ] = P[ she is actually suffering from the “heartbreak of psoriasis” and positive result ] + P[ she is not suffering from the “heartbreak of psoriasis” and positive result ]

P[ she is actually suffering from the “heartbreak of psoriasis” and positive result ] = P[ test used will correctly label a person who is suffering from the “heartbreak of psoriasis” as a sufferer ] *P[ suffering from the “heartbreak of psoriasis” ]

P[ she is actually suffering from the “heartbreak of psoriasis” and positive result ] = 0.9*0.05 = 0.045

P[ she is not suffering from the “heartbreak of psoriasis” and positive result ] = P[ will incorrectly label a person who is not suffering from the “heartbreak of psoriasis” as being a sufferer ] *P[ not suffering from the “heartbreak of psoriasis” ]

P[ she is not suffering from the “heartbreak of psoriasis” and positive result ] = 0.95*0.4 = 0.38

P[ positive result ] = 0.38 + 0.045 = 0.425

P[ she is actually suffering from the “heartbreak of psoriasis” | positive result ] = P[ she is actually suffering from the “heartbreak of psoriasis” and positive result ]/P[ positive result ]

P[ she is actually suffering from the “heartbreak of psoriasis” | positive result ] = 0.045/0.425

P[ she is actually suffering from the “heartbreak of psoriasis” | positive result ] = 0.1059

P[ she is actually suffering from the “heartbreak of psoriasis” | positive result ] = 10.59%

I think you used 60% instead of 40% that is why it's giving you a wrong answer