Number of probability distributions on a finite set Is the cardinality finite (of the set of probability distributions on a finite set $A$)? Countable? Uncountable?

I think finite gets rule out because the probability distribution on a 2 element set can assign any probability in $[0,1]$ to the second element, so there should be an uncountable number of probability distributions? 

This answer Say a probability distribution  over reals has cardinality $2^{\mathbb{N}_0}$.  So it is indeed uncountable?
(Edited in response to comment) Answer is: Uncountable if the set has cardinality of at least two, as shown in the example in the question. If the set has cardinality 1 then there is only one possible probability distribution.
(Edited in response to comment) Answer is: Uncountable if the set has cardinality of at least two, as shown in the example in the question. If the set has cardinality 1 then there is only one possible probability distribution.