Here my focus is mainly on $Ord$ and questions such as:
An ordinal is chosen by random. What is the probability of the event that it is a cardinal number?
A coin is tossed proper class many times. What is the probability of an event in which the first $\alpha$ tosses have resulted in tail?
In some sense these questions are similar to the following questions which take place in sample spaces $\mathbb{N}$ and $\{0,1\}^{\mathbb{N}}$ respectively:
A natural number is chosen by random. What is the probability of the event that it is a prime number?
A coin is tossed (countably) infinite times. What is the probability of an event in which the first $n$ tosses have resulted in tail?
The difficulty is that $Ord$ is a proper class and it is not immediately clear what type of object should the $\sigma$-algebra of events be. It seems formalizing the notion of a probability space for a proper class sample space needs some works and stronger axiomatic systems than ZFC.
Question: How to formalize the notion of probability space for proper class sample spaces? Are there any references for development and use of theory of probability for proper class spaces?
Under the usual product measure on $2^{ORD}$ (whatever that may mean), the probability that the first $\alpha$ coin tosses be tail is $2^{-|\alpha|}$ which is zero unless $\alpha$ is finite. I cannot make sense of your first question.