I'd like help correctly applying Bayes Rule to an unusual placebo-controlled trial design, which had users guess whether they were in the placebo or treatment groups.
So, in a normal Placebo trial, the probability that a person gets better given that they were given the treatment is
P(T|B) = (P(T) * P(B|T)) / P(B)
In the study I'm analyzing, participants were also asked to guess whether they were part of the Placebo or Treatment groups. Some guess correctly. Some do not.
Here's the question I'm trying to answer: What is the probability that an individual gets better but wrongly guesses they were in the treatment group? That is, they got better, but got the placebo.
To use Bayes rule, do I need to add subgroupings? Because we know that some people feel better and also correctly guessed they were given a placebo. So, not everyone in the placebo group is being fooled. It feels like they should be left out (or accounted for). And, if so, what is the correct way to apply that in Bayes?
So, the total groups and permutations we have are: B=Better N=No effect/worse T=Treatment P=Placebo GR=Guess right GW=Guess wrong
Thanks for any help you can give to this problem and let me know if I can make it more clear!