If $q \geq \delta$ then $P[Z_8 \geq \delta^8] \leq 10^{-2}$

Question

Consider a sequence of independently and identically distributed random variables $X_1,X_2,...,X_n$, where $X_i \in \mathcal{X} = \{ 0,1\}$ and $P[X_i = 1]=q$ for all $i \in \{ 1,...,n \}$. Let $\delta$ be a real-valued constant, $\delta \in (0,1]$. Let us define the random variable: $Z_n = \prod_{i=1}^{n} X_i$.

Is the following statement correct? If $q \leq \delta$, then $P[Z_8 \geq \delta^8] \leq 10^{-2}$.

I know that $\delta$ is a real-valued constant between 0 and 1 (without including 0), therefore $P[X_i \geq \delta]=P[X_i=1]=q$. However, I am not sure how to proceed from here.

Answer 1

Since $X_i$'s are $0-1$ valued and $1\geq \delta>0$, $Z_8 \geq \delta^8$ if and only if $X_1,X_2,\ldots X_8$ are all equal to one. Therefore

$$ P[Z_8 \geq \delta^8] = \prod_{i=1}^8P[X_i=1] =q^8. $$ Now your statement is correct as long as $q^8 \leq 10^{-2}$.