# Bayes, and the Horse He Rode In On

#### raywood

##### New Member
A professor presented this question: "If a cancer screen comes back positive in 68 of 75 cases of actual cancer and in 5 of 191 cases of no cancer, when the actual cancer rate in the total sample is 1/3, what is the probability of a false negative test result (patient has cancer but test comes back negative)?"

I was thinking the simplest calculation is as follows: the cancer screen identifies 68/75 = 91% of cases where the patient has cancer. It fails to identify 7/75 = 9% of cases that it should identify. The probability of a false negative is 9%.

One person tells me that it is necessary to use Bayes' Theorem to arrive at a correct answer. I don't know how to do that. He says the correct answer is 4.57%. But when I use the Bayesian calculator at http://www.vassarstats.net/clin1.html, I don't get that number.

Possibly I am using the calculator wrong. Even so, I don't know why the 9% figure wouldn't be right. I am wondering whether this is a situation where Bayesian and non-Bayesian statisticians diverge.

In case anyone is wondering, this is not a homework question. It used to be a quiz question, but right now it's just spilt milk.

Thanks for any insights.

#### Dason

I am wondering whether this is a situation where Bayesian and non-Bayesian statisticians diverge.
No because this is just an example of using Bayes theorem. It's straight probability and frequentists and bayesians both have no problem with this.

#### Mean Joe

##### TS Contributor
I was thinking the simplest calculation is as follows: the cancer screen identifies 68/75 = 91% of cases where the patient has cancer. It fails to identify 7/75 = 9% of cases that it should identify. The probability of a false negative is 9%.
Write it out, you made a common mistake.
P{positive test | cancer} = 68/75,
therefore
P{negative test | cancer} = 7/75.
This is what you actually calculated. But the question of course is asking P{cancer | negative test}.

It is a very common mistake; see the example(s) in this article -- I myself just read the first half: http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/