I am stuck with something that looks very simple but I am not able to find where I am wrong. Let $\xi_k$ with $k=1,...,n$ be $n$ iid Bernoulli random variables such that
$$ \mathbb{P}\left[\xi_k=1\right]=p,~~\mathbb{P}\left[\xi_k=0\right]=1-p. $$
The product
$$ \psi = \xi_1\,\xi_2\,\cdot\cdot\cdot\xi_n $$
is stil a Bernoulli variable with either $\psi=1$ or $\psi=0$. Clearly $\mathbb{P}\left[\psi=1\right]=p^n$, therefore $\mathbb{P}\left[\psi=0\right]=1-p^n$. Where I am wrong if I say
$$ \mathbb{P}\left[\psi=0\right]=\mathbb{P}\left[\{\xi_1=0\}\cup\{\xi_2=0\}\cup ...\cup\{\xi_n=0\}\right]=n\,\left(1-p\right) ? $$
We have that $$ \Pr\{A\cup B\}=\Pr\{A\}+\Pr\{B\}-\Pr\{A\cap B\}=\Pr\{A\}+\Pr\{B\}-\Pr\{A\}\Pr\{B\} $$ if $A$ and $B$ are independent events. For example, if $n=2$, \begin{align*} \Pr\{\psi=0\}&=\Pr(\{\xi_1=0\}\cup\{\xi_2=0\})\\ &=\Pr\{\xi_1=0\}+\Pr\{\xi_2=0\}-\Pr\{\xi_1=0\}\Pr\{\xi_2=0\}\\ &=1-p^{2}. \end{align*} In general, we need to use the inclusion–exclusion principle.