I tried searching everywhere but couldn't find a formalization of this.
I guess the set of possible outcomes $\Omega$ is the set of all sequences of zeros and ones, where $1$ at the $n$th position means that the monkey typed Hamlet correctly at its $n$th attempt, and $0$ otherwise. The $\sigma$-algebra $\mathcal{M}$ on $\Omega$ could be the one generated by the sets $A_n$, where $A_n$ is the set of all sequences with $1$ in the $n$th position.
How do we define a probability measure $\mathbb{P}$ on $(\Omega, \mathcal{M})$ such that $\mathbb{P}(A_n) = c > 0$, where $c$ is the probability of the monkey typing it correctly in one attempt?