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?

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.
