  A fruitful way of obtaining meaningful, possibly concrete, algorithmically
random numbers is to consider a potential behaviour of a Turing machine and its
probability with respect to a measure (or semi-measure) on the input space of
binary codes. For example, Chaitin's Omega is a well known Martin-Loef random
number that is obtained by considering the halting probability of a universal
prefix-free machine. In the last decade, similar examples have been obtained
for higher forms of randomness, i.e. randomness relative to strong oracles. In
this work we obtain characterizations of the algorithmically random reals in
higher randomness classes, as probabilities of certain events that can happen
when an oracle universal machine runs probabilistically on a random oracle.
Moreover we apply our analysis to different machine models, including oracle
Turing machines, prefix-free machines, and models for infinite online
computation. We find that in many cases the arithmetical complexity of a
property is directly reflected in the strength of the algorithmic randomness of
the probability with which it occurs, on any given universal machine. On the
other hand, we point to many examples where this does not happen and the
probability is a number whose algorithmic randomness is not the maximum
possible (with respect to its arithmetical complexity). Finally we find that,
unlike the halting probability of a universal machine, the probabilities of
more complex properties like totality, cofinality, computability or
completeness do not necessarily have the same Turing degree when they are
defined with respect to different universal machines.
