The probability that the product of two elements of finite group algebra is zero

Let ${\mathbb{F}}_{q}G$ be a finite group algebra. We denote by $P({\mathbb{F}}_{q}G)$ the probability that the product of two elements of ${\mathbb{F}}_{q}G$ be zero. In this paper, the general formula for computing the $P({\mathbb{F}}_{q}G)$ are established for the cyclic groups $C_n$, the Quaternion group $Q_8$ and the symmetric group $S_3$, for some cases.