Why $P(C=c)$ instead of simply $P(C)$ For example, reading a representation of Bayes Theorem 
$$P(C=c|E) = P(C=c)P(E|C=c) / P(E)$$
Let's use the common example of $P(C)$ = probability of cancer and $P(E)$ = probability of positive mammogram test.  Why denote it as $P(C=c)$ and what does that represent?
 A: It really depends on the definition of $C$. 


*

*If $C$ is defined as an event, then use $P(C)$. 

*If $C$ is defined as a random variable that represents, say, the categories of diseases, then use $P(C = c)$, with the understanding that it essentially means $P(\{C = c\})$, where $\{C = c\}$ is an event. (More technically, you must think of an underlying probability space $(\Omega, \mathscr{F}, P)$ on which the random variable $C$ is defined so that $\{C = c\} := \{\omega: C(\omega) = c\}$ is a member of the $\sigma$-field $\mathscr{F}$. Here, $\mathscr{F}$ can be thought as a collection of events that we can measure their uncertainties, i.e., assign probabilities.) 
Based on your information, I am inclined to agree with you that $C$ is the event of getting cancer. Therefore $P(C)$, instead of $P(C = c)$ is the accurate notation here. My guess is also based on a notation convention (though not strictly) in probability: people tend to use initial Latin letters $A, B, C, D, E$ to represent events, while use bottom Latin letters $X, Y, Z$ to represent random variables. 
A: I think people are sloppy on the notation $P(C)$. $C$ is a random variable and it can have some values, say $\{1, 2, 3\}$. 
The formal notation should be $P(C=c)$ where $c \in \{1,2,3\}$. $P(C)$ is a distribution, i.e., table (in discrete case), say 
\begin{cases} 
      P(C=c)=0.2 & c=1 \\
      P(C=c)=0.3 & c=2 \\
      P(C=c)=0.5 & c=3 \\
   \end{cases}
But not a probability number.
