Does this probability problem make sense? I just got a probability problem as follows.


  Every person in a region has a probability of getting infected by HIV
  of 0.1. If the total population of that region is 1000 persons, find
  how many person approximately get infected?


What I don't understand is how the probability for each person correlate to the population. Is this problem correct? I don't know  which concept to be applied here. Independent event, mutually exclusive event, etc,... I am really confused!
There is some ambiguity, but I assume the question intends for you to see that with probability 0.1, a person is infected by HIV where the probability of each person getting infected is independent of each other. So the number of people infected is the expected value of a binomial distribution where $n = 1000$ and $p = 0.1$. Then the answer is simply $np = 100$. 

Another way to see this is to define a random variable $X_{i}\sim Ber(0.1)$ where $X_{i} = 1$ if the $i$th person is infected or $0$ otherwise. Then by linearity of the expected value,

$$\mathbb{E}\left(\sum_{i=1}^{1000}X_{i}\right) = \sum_{i=1}^{1000}\mathbb{E}(X_{i}) = \sum_{i = 1}^{1000}0.1 = 1000(0.1)$$
You can use the Central limit theorem to do the approximation.
Central Limit Theorem