`  == nature makes its move ==  :Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ω, from the sample space Ω. Then we say that all events from  containing the selected outcome ω (recall that each event is a subset of Ω) “have occurred”. The selection performed by nature is done in such a way that if we were to repeat the experiment an infinite number of times, the relative frequencies of occurrence of each of the events would have coincided with the probabilities prescribed by the function P.  This seems very muddledis there a more formal description of what happens here?  For example, if I repeatedly flip an unbiased coin (infinite Bernoulli process) and ``nature`` selects the sequence H,T,H,T,H,T... then the frequencies of occurrence coincide with the expected 50% heads, but I wouldn't call it random.  Thanks.   `