## Use of Bayes Theorem

Hi, I am new here, thanks for having me. I am not a statistician though I have done a stats course at Uni (a long long time ago!).

I am trying to understand how Bayes Theorem can be applied, and have set up a hypothetical.

Suppose a cancer treatment has been tested and found to have a 30% success rate 1 year after treatment. Suppose now we want to see if the test results are working out in practice, so we ask all hospitals who are using this treatment to test their patients one year after treatment, using particular testing equipment and report their results. The testing equipment has a known rate of 20% false negatives and 10% false positives in finding cancers.

The plan is to run Bayes Theorem every time a new result comes in, to update the estimate of the success rate. The hypothesis (h) is that the cancer treatment continues to be successful. The prior probability Pbefore = 0.3. When a cancer-free result comes in, the probability of this result if the hypothesis is true Ph= 0.8, and if the hypothesis is false P~h=0.2. And when a positive cancer result comes in, Ph = 0.1 and P~h = 0.9.

So the first result is that the patient is cancer free, so I do the Bayes calc and get Pafter = 0.63 (if my calcs are correct). Then the next result is a patient who has cancer after 1 year, so the calc is done again with Pbefore = 0.63, and the result is Pafter = 0.16. And so on as new results come in.

This is just a hypothetical to test my understanding of Bayes, and may not be medically sensible, but can anyone tell me please:

1. Is this the sort of situation where it is legitimate to use Bayes Theorem?
2. Is there a better way to do the calculation?
3. Have I done the numbers right (the result seems to change around a lot)?

I hope this isn't too much to ask. Thank you.