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?


2 Answers 2


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.

  • $\begingroup$ If $C$ is defined as an event, then you need to specify what distribution do you mean by $P$. $\endgroup$ Dec 20, 2017 at 23:16
  • 2
    $\begingroup$ 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, 2017 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, 2017 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.

  • 1
    $\begingroup$ This is right, but perhaps you can clarify what {1,2,3} would mean in the cancer example. $\endgroup$
    – dimitriy
    Dec 20, 2017 at 21:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.