Definition of conditional probability for the intersection of 2 events - am I right? Am I right saying that (assuming that $P(A)&gt;0, P(B)&gt;0, P(C)&gt;0$)

$P(A\cap B|C)=\frac{P(A\cap B\cap C)}{P(C)}?$
Yes, you are correct. To help see for yourself, let's call the event $A \cap B$ by a new name, $E$. The definition of conditional probability gives us that
\begin{equation*}
P(E \mid C) = \frac{P(E \cap C)}{P(C)}
\end{equation*}
We also know that $E \cap C = (A \cap B) \cap C = A \cap B \cap C$, as events, so we get the result you ask about.

(Also, note that we only need to assume $P(C) &gt; 0$. The result is perfectly valid when $P(A) = 0$ and/or $P(B) = 0$.)
This is comes from defination of conditional probability for two sets, you can also try for more than three and any finite as well infinite numbers of sets.as i think well?