# 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

Letâ€™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?