\problemdata                                                             
        {Mutual independent events - proof} % latex-friendly label for the prob.
        {Independence, probability} % The topic(s) of the problem content.  
        {from From ``A first course in Probability" p 119, exercise 6} % Source (if known)                                          
        {S01 Final, 9} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Prove that if $E_1, E_2,\ldots,E_n$ are mutually independent events, then
\[
\Pr\left[E_1\union E_2\union \cdots \union E_n\right]=1-\prod^n_{i=1}\left[1-P(E_i)\right]
\]

\solution{
  \begin{eqnarray*}
    \Pr\left[E_1\union E_2\union \cdots \union E_n\right] &=&
    1 - \Pr\left[\overline{E_1\union E_2\union \cdots \union E_n}\right] \\
    &=&
    1 - \Pr\left[\overline{E_1} \intersect \overline{E_2} \intersect
      \cdots \intersect \overline{E_n}\right] \\
    &=&
    1 - \prod_{i=1}^n\Pr\left[\overline{E_i}\right] \\
    &=&
    1 - \prod_{i=1}^n\bigl(1 - \Pr[E_i]\bigr).
  \end{eqnarray*}
  
  The most common error here was to claim that since the $E_i$ are
  independent, $\Pr(E_1 \union \cdots \union E_n) = \Pr(E_1) + \cdots
  + \Pr(E_n)$. This is false, it holds when the events are disjoint.}

\end{problem}
                                 