Bayes Theorem

A first look at Bayes

Bayesian reasoning is about how to revise our beliefs in the light of evidence.

A nurse is screening a set of students for Diseasitis. From past studies we know that 20% of students will have Diseasitis at this time of the year. A color-changing tongue depressor, is used to identify who has Diseasitis and not. Among patients who have Diseasitis, 90% turn the tongue depressor black. It also turns black 30% of the time for healthy students.

One of the students comes into the nurses office and takes a test and turns the tongue depressor black. what is the probability that he has Diseasitis.

I have read this article quite some time back and remember the solution. It seems easier to do it the following way as suggested in the same article on Arbital.

Glossary

No. of students who have Diseasitis = N_D

No. of students who do not have Diseasitis = N_dnD

No. of students whose tongue depressor turns black & have Diseasitis = N_D_B

No. of students whose tongue depressor turns black & do not have Diseasitis = N_dnD_B

Action

Let’s say there are 100 students.

20% of students will have Diseasitis.

$ N_D = 20 

$ N_dnD = 80 

Among patients who have Diseasitis, 90% turn the tongue depressor black.

$ N_D_B = 90% of N_D = 18

It also turns black 30% of the time for healthy students.

$ N_dnD_B = 30% of N_dnD = 24

One of the students comes into the nurses office and takes a test and turns the tongue depressor black. what is the probability that he has Diseasitis.

P (a student who has turned the depressor black has Deseasitis): $ (N_D_B)/(N_D_B + N_dnD_B) = 18/(24+18) = 18/42 = 3/7 = 43%

Open Issue

Why and where is Bayes theorem useful for us?