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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?

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It really depends on the definition of $C$.

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

  2. 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 an adequate 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.

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  • $\begingroup$ If $C$ is defined as an event, then you need to specify what distribution do you mean by $P$. $\endgroup$ – Jakub Bartczuk Dec 20 '17 at 23:16
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    $\begingroup$ On the contrary, we first define $P$. The term "distribution" is only relevant if we have additionally defined a random variable. If we are only talking about events, no need to introduce distribution. Again, $(\Omega, \mathscr{F}, P)$ is the root, things like "random variables, distributions" are derivatives. $\endgroup$ – Zhanxiong Dec 20 '17 at 23:46
  • $\begingroup$ Nice answer (+1), however given the fact that it asks about very basic problem, it would seem to be reasonable to add an example that illustrates it besides the formal definitions. $\endgroup$ – Tim Dec 21 '17 at 8:42
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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.

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    $\begingroup$ This is right, but perhaps you can clarify what {1,2,3} would mean in the cancer example. $\endgroup$ – Dimitriy V. Masterov Dec 20 '17 at 21:19

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