Probability that an arbitrary element of a field equals a uniformly random value

Question

Hypothesis: we define all elements over a finite field: $\mathbb{F}_p$, where $p$ is a large prime number.

We draw $r_i$ uniformly at random from the field.

Question: What is the probability that a fixed element of the field $\alpha$ equals $r_i$?

Answer 1

There is just one field ${\mathbb F}_p$ with $p$ elements, called the prime field belonging to $p$. In addition there are for all $n\geq1$ finite fields ${\mathbb F}_q$ with $q=p^n$.

Anyway: If you have a field ${\mathbb F}_q$ with $q$ elements then the uniform distribution assigns the probability ${1\over q}$ to each individual element, in particular to the element $\alpha$ you have in your head. The answer to the question therefore is ${1\over q}$.

If you choose $\alpha$ according to some "secret" distribution $(P_k)_{1\leq k\leq q}$ then the probability of a match with the element $r$ chosen uniformly by your enemy comes to $$\sum_{k=1}^q P_k\>{1\over q}={1\over q}$$ as well.