# 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?

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 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.

• If $C$ is defined as an event, then you need to specify what distribution do you mean by $P$. – Jakub Bartczuk Dec 20 '17 at 23:16
• 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. – Zhanxiong Dec 20 '17 at 23:46
• 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. – Tim Dec 21 '17 at 8:42

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.

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