What is the practical meaning of probability vectors?

Question

I have been reading a lot about probability vectors, as a part of "Introduction to Probability" course. Now, whenever it was mentioned, it was defined theoretically as a vector whose entries add up to 1 (and are not negative.). Additional or other properties described were too ambiguous or complex for me. Sources in my own language haven't made it clear either. My educatively guessed notion of that would be: The probability vector $(a_1,a_2,...,a_n)$ in a space $\Omega=\{A_1,...,A_n\}$ represents an initial state in which $p(A_i)=a_i$. In particular, a stationary stationary probability vector is an initial state that will not be affected by multiplication by any power of the respective stochastic matrix. Is what I said correct or close? If it isn't, what does the vector stand for? When looking at solutions, the solver reads all sort of things into the vectors they examine or arrive at, and I can't possibly know why, for they wouldn't go over the definitions or basic theorems which they consider an obvious truth. I would appreciate your help.

Answer 1

That's correct. Some comments:

• This notion is not attached to stochastic processes. You can use a probability vector to represent a discrete distribution in a context which involves no time dependence.
• In general we measure the probabilities of events, which are subsets of the sample space, rather than elements of the sample space. Traditionally we name events with capital letters. So it would be a bit better to say $\Omega = \{ b_1,\dots,b_n \}$ and $P(\{ b_i \})=a_i$.