The exercise of my homework assignment says
"Let $E$, $F$, $G$ be independent events with probability $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{5}$. Let $X$ be the number of events that occur. Determine the random variable $X$, its mean value and variance."
My concern is on the first question, and I hope to be able to explain it. Shouldn't I know how many of the event that occur belong to $E$, $F$, $G$? On a first guess I expect not to be able to write analitically $\Bbb{P}(X=k)$. My only attempt is by using conditioning, but I guess I should know how many of the three events occurred; defining $X\colon = X_1+\cdots+X_n$
\begin{multline} \Bbb{P}(X = k) = \Bbb{P}(X=k|X_1,\cdots,X_{n_1}\in E)\cdot\Bbb{P}(X_1,\cdots,X_{n_1}\in E) + \\ \Bbb{P}(X=k|X_{n_1+1},\cdots,X_{n_2}\in F)\cdot\Bbb{P}(X_1,\cdots,X_{n_2}\in F) + \\ \Bbb{P}(X=k|X_{n_2+1},\cdots,X_{n_3}\in G)\cdot\Bbb{P}(X_1,\cdots,X_{n_3}\in G) + \\ \Bbb{P}(X=k|X_{n_1 + n_2 + n_3},\cdots,X_{n}\notin E\cup F\cup G)\cdot\Bbb{P}(X_{n_1 + n_2 + n_3},\cdots,X_{n}\notin E\cup F\cup G) \end{multline}
where $n_1,\,n_2,\,n_3\,$ are the respective numbers of the events occurring that belong to $E$, $F$, $G$ respectively.
Let me know if my question is as unclear as the exercise is to me.
