# Bayes Theorem - counter intuition (2)

#### Counter-intuitive

In [yesterdays post] we saw the following question:

20% of a population of students have Diseasitis. 90% of those who have Diseasitis, turn a tongue depressor test black. 30% of those who do not have Diseasitis, turn the same tongue depressor test black.

Turning black according to the test means that one has Diseasitis. The test is 90% of the time right for those that have Diseasitis. The test is 70% of the time right for those that do not have Diseasitis. This stems from the fact that 30% of the time the test wrongly predicts that people who do not have Diseasitis, have Diseasitis.

The question to really answer is that if the tongue-depressor-test is usually trust worthy. In other words, is the tongue-depressor-test, going to give me the right results >50% of the time I use it.

Great. We would think it right to say that atleast 70% of the time the tongue-depressor-test gets things right as seen with the 90% and 70%.

But yesterday, we saw that out of the people who take the tongue-depressor-test and got it black, only 43% would actually have Diseasitis. That is to say that we were initially wrong when we said atleast 70% of the time the tongue-depressor gets things right.

Initially before the test, a nurse assumes that there is a 20% chance that a general population of students have Diseasitis. With the tongue-depressor-test showing black, the probability of someone being sick is 43%, i.e, higher. The test is useful in the fact that it is better than no test, but it’s still not good enough, rather it’s not conclusive enough.

#### Follow-up question

Our ultimate interest is in Diseasitis, and people having them. So we explore some more.

What’s the probability that a student who does not turn the tongue depressor black - a student with a negative test result - has Diseasitis?

```
$ N_D = 20
$ N_dnD = 80
```

Among patients who have Diseasitis, 10% do not turn the tongue depressor black - negative test result.

```
$ N_D_dnB = 10% of N_D = 2
```

It also gives negative results (does not turn black) 70% of the time for healthy students.

```
$ N_dnD_dnB = 70% of N_dnD = 0.7*80 = 56
```

The probability that a student who does not turn the tongue depressor black and has Diseasitis:

```
$ N_D_dnB / (N_D_dnB + N_dnD_dnB) = 2/58 = 3.5%
```

It appears to me that these are the two numbers we need to see, when someone claims a test can predict:

1) The probability that a student who does not turn the tongue depressor black and has Diseasitis: 3.5 %

2) The probability that a student who turns the tongue depressor black and has Diseasitis: 43%

Awesome. Next!

#### P.S

All credit goes to Arbital for the beautiful teaching.

#### Open Issue

Why and where is Bayes theorem useful for us?