How to calculate if probability is negligible or not If i have probability $Q = 2C(A\times B)$ where $A$ and $B$ are unknown probabilities and $C$ is a non-negligible probability, what can i speculate about probability $Q$ and how can i calculate bounds on probabilities $A$ and $B$ in order to make this value non-negligible?
First of all, keep in mind that all meaningful probabilities must be between $0$ and $1$.  In particular, this means that, if $X$ and $Y$ are probabilities, $0 \le XY \le \min(X, Y)$.

In cryptography, a probability is considered "negligible" if it is very small.  What actually counts as "negligible" depends on context, but typically, we're talking about probabilities on the order of $1/2^{128}$ or less.

Conversely, a "non-negligible" probability may be high, perhaps even close to $1$.  In fact, since we typically want probabilities (of an attack succeeding) to be small, we may safely assume that, in the worst case, all non-negligible probabilities are $\approx 1$.  This leads to the following "rules":


The product of two negligible probabilities is (always) negligible.
The product of a negligible and a non-negligible probability is also (always) negligible.
The product of two non-negligible probabilities is (usually) not negligible.


Corollary: For the product of several probabilities to be negligible, at least one of them must (usually) be negligible.

Multiplying a probability by a constant $\approx 1$ (say, between $1/2^{16}$ and $2^{16}$) does not (usually) change its negligibility.
The sum of two negligible probabilities is (usually) negligible.
The sum of a negligible and a non-negligible probability (or of two non-negligible probabilities) is never negligible (and might not always, strictly speaking, be a probability).


Thus, without knowing anything about $A$ and $B$, except that they are probabilities, the most we can say about $Q = 2C(AB)$ is that it is somewhere between $0$ and $2C$.  Furthermore if $C$ is not negligible, then $2C$ is definitely not negligible; thus, for $Q$ to be negligible, $AB$ must (generally) be negligible.

For $AB$ to be negligible, it's sufficient for either $A$ or $B$ to be negligible.  If neither $A$ nor $B$ is negligible, their product cannot be assumed to be negligible either &mdash; although it's possible for $AB$ to be negligible (e.g. $AB \approx 1/2^{128}$) even if both $A$ and $B$ are only "nearly negligible" (e.g. $A \approx B \approx 1/2^{64}$).

Conversely, for $Q = 2C(AB)$ not to be negligible, neither $A$ nor $B$ may be negligible.  (Strictly speaking, for any specific threshold of negligibility, the factor of $2 &gt; 1$ could bump a probability just below the line to just above it; but the whole point of using terms like "negligible" is to note that we're speaking loosely and usually ignoring such factors of $\approx 1$.)  It's still possible for all of $A$, $B$ and $C$ to be above our negligibility threshold, but for their product to be below it (if two or more of them are "almost negligible"), but we cannot assume this without knowing their values more precisely.