Formal language for describing variability of a random experiment

Question

I have a random experiment with its probability space $(\Omega, \cal{F}, P)$ but no random variable defined over it. Is there any formal language for describing the variability of the random experiment? E.g. I feel the following three experiments have increasing variability, but I am not sure how to formalize this feeling (or if you have a different feeling, how to formalize that):

1. $\Omega=\{\omega_1\}$, $\quad\ $ $Pr(\omega_1)=1$
2. $\Omega=\{\omega_1,\omega_2\}$, $Pr(\omega_1)=0.9$, $Pr(\omega_2)=0.1$
3. $\Omega=\{\omega_1,\omega_2\}$, $Pr(\omega_1)=Pr(\omega_2)=0.5$

I am not opposing to the idea of introducing a random variable if that helps. I am just saying there is none in the original setup.

Answer 1

A salient candidate is entropy. For a random experiment $X$ with possible outcomes $\omega_1,\dots,\omega_n$ and the corresponding probabilities $P(\omega_1),\dots,P(\omega_n)$, entropy is defined as $$ H(X) =-\sum_{i=1}^n P(\omega_i) \ln \left( P(\omega_i) \right). $$ It measures the average level of "information", "surprise", or "uncertainty" inherent in the experiment's possible outcomes.

_{The fact that Wikipedia's entry on entropy defines entropy w.r.t. a random variable rather than the underlying experiment can be considered a failure. According to @whuber,}

_{This is an unfortunate case where Wikipedia fails us. Entropy most fundamentally is a property of a discrete probability measure. Everything else is derived from that concept.}