I am trying to derive a lower bound of $2^k$ for the size of the sample space $\Omega$ given that in the probability space there exists $k$ non-trivial independent events. I am not looking for a solution per se, but rather a hint.
I suppose I don't fully understand the connection between the size of an event and the surrounding sample space. Namely I do understand that the fact that the events $E_1, \dots, E_k$ are non trivial mean that $\mathrm{Pr}(E_i) \in (0, 1)$ or equivalently that each event contains at least one element, but no event contain all elements of the space. But how do you go about deriving the lower bound from this? To me $2^k$ speaks that some element has a binary choice to be or not to be included in all of the $k$ events in the probability space. $2^k$ could also mean the number of possible subsets for $k$ elements. However I don't see how either of these reasoning strategies yields anything useful, as they don't take into account the relationships between the events $E_1, \dots, E_k$.