Random variable and trying to find $E(N)$ 
Given
$N = \text{random variable that counts the fraction of trials that are successful
trials} = 12$
$N = S/12$
$S = \text{number of successful trials}$
$E(N)= ?$
i don't know how to find $E(N)$ is there a specific formula? can someone help me out with this problem
 A: It is useful to introduce some notation. Define random variables $X_1,X_2,\dots, X_n$ by $X_k=1$ if the $k$-th trial is successful, and $X_k=0$ if the trial is not successful. 
Then  $S=X_1+X_2+\cdots +X_n$: the sum of the $X_i% counts the number of successful trials.
Finally, note that  $N=\dfrac{S}{n}$: $N$ is the proportion of successful trials. Then 
$$E(S)=E(X_1+X_2+\cdots +X_n)=E(X_1)+E(X_2)+\cdots +E(X_n).$$  
We calculate $E(X_i)$. The probability that $X_i=1$ is the probability of $2$ heads, which is $\frac{1}{4}$. So $E(X_i)=(1/4)(1)+(3/4)(0)=1/4$.
It follows that $E(S)=\dfrac{n}{4}$. 
Note that $E(N)=E\left(\dfrac{1}{n}S\right)=\dfrac{E(S)}{n}$.
Thus $E(N)=\dfrac{1}{4}$.
Remark: The answer is intuitively clear. Actually, the number of successes in $n$ trials has binomial distribution with "$p$", the probability of success, equal to $1/4$.  The proportion of successes should have expected value $\frac{1}{4}$. We introduced the random variable machinery because it will be long run useful.
