What do Commas Mean in Probability

Question

For example in my textbook I am doing conditional probabilities and there are commas in the conditional. Like P(A,B|C). What does this comma do in the math equation? A simple example would also really help, thanks!

Answer 1

It's common to use commas to denote set intersection. That is, $P(A,B) = P(A \cap B)$. One also sometimes sees this written simply as $P(AB)$. In natural language, $P(A,B \mid C)$ becomes something like "the probability of A and B given C."

As a simple example, take the uniform distribution on $\{ 1,...,10 \}$. You could interpret this as rolling a fair 10-sided die. Let $A = \{1,2,3,4,5 \}$ (the die comes up less than 6), let $B = \{ 1,3,5,7,9 \}$ (the die shows an odd number), and let $C = \{ 3,6,9 \}$ (the die shows a multiple of 3).

Then, using the definition of conditional probability,

$P(A,B \mid C) = P(A \cap B \mid C) = \frac{P(A \cap B \cap C)}{P(C)} = \frac{P(\{3 \})} {P(C)} = \frac{1/10}{3/10} = \frac{1}{3}$.

We have calculated the probability that the die shows an odd number less than $6$, given that it shows a multiple of 3.