
6 PRELIMINARIES
that the probability a coin lands on Heads is θ , then the understanding is that, in an
infinite sequence of coin tosses, the limiting proportion of Heads observed equals θ.
Mathematically, this can be written as
θ = lim
n→∞
|{i = 1,... ,n : X
i
= 1}|
n
,
where X
1
,.. . ,X
n
are independent binary random variables, where X
i
= 1 corresponds
to the coin landing on Heads, and |B| denotes the cardinality of a finite set B. It is
straightforward to extend this idea to any other setting, not just Bernoulli trials. The
other interpretation is a subjective one, and we call this belief probability. In this
case, the belief probability assigned to an event ...