the number of components working at a particular time Suppose a system has $10$ components and that a particular time the $j$'th component is working with probability $1/j$ for $j=1,2,\dots,10$. How many components do you expect to be working at that particular time?
I have only found the probability of the components and I know that the sum of the possible probabilities is $1$.
How do I go about solving this question?
 A: We use Indicator Random Variables. Let $X_i=1$ if component $i$ is working, and let $X_i=0$ otherwise. Then the number $N$ of components working is a random variable given by
$$N=X_1+X_2+\cdots+X_{10}.$$
By the linearity of expectation,
$$E(N)=\sum_{1}^{10}E(X_i).$$
But $E(X_i)=\dfrac{1}{i}$. Add up, $i=1$ to $10$.
Remark: Note that expectations add even if the random variables are not independent.  
The method of indicator random variables can be extremely useful in the computation of means, and often of variances. It can, as in this case, be used to compute the mean of a random variable like our $N$ without finding the distribution function of that random variable.
A: The expectation that the $j$'th component is working at that moment is $1/j$.
So the total expectation of components working is the sum of the expectations.
Components working $=\sum_{j=1}^{10} 1/j$.
A: Expectations are additive.  This is the single most important thing to know about expectations.
So imagine that component 1 is the only component, and find the expected number of working components at any time.
Then imagine that component 2 is the only component, and find the expected number of working components at any time.
Do that for each of the ten components, and add up the ten expected numbers you get, and that is the answer for all of them at once.
