EDITED THANKS TO THE COMMENTS BELOW
Consider a random variable $C$ distributed as a Poisson with rate $\lambda$ (see formulas here). Further, assume that the rate $\lambda$ is distributed as a Gamma with mean $1$ and variance $\phi$.
First, note that the probability that $C$ is at least equal to $1$ conditional on $\lambda$ is $$ P(C\geq 1|\lambda)=1-P(C=0|\lambda)=1-\exp(-\lambda) $$
Second, note that the unconditional probability that $C\geq 1$ is (thanks to the comments below) $$ P(C\geq 1)=1-\int_{t} \exp(-t) f(t) dt $$ where $f$ is the Gamma distribution with mean $1$ and variance $\phi$.
Third, I believe that for any $c\geq 1$ $$ P(C\geq 1| \lambda, C=c)=P(C\geq 1| C=c)=1 $$ while $$ P(C\geq 1| \lambda, C=0)=P(C\geq 1| C=0)=0 $$
Question: a book that I'm using considers the following objects $$ (A)\text{ }1-\exp(-E(\lambda)) $$ and $$ (B)\text{ }1-\exp(-E(\lambda| C)) $$ where $E$ denotes the expectation. In particular, B is referred to as Bayesian Credibility Premium.
What do A and B represent? What is the relation between A and B and the objects derived above? How do I compute $E(\lambda|C=c)$ for any $c$ given only the information above?