medium Probability
A release-checked medium problem for training Combinations and Permutations, Conditional Probability.
Suppose that $\Omega = \{1,2,\dots,p\}$, where $p$ is a prime integer. One $\omega \in \Omega$ is selected uniformly at random. Find the number of pairs of subsets $A, B \subseteq \Omega$ such that the events $\{\omega \in A\}$ and $\{\omega \in B\}$ are independent events. Find this when $p = 7$.
This Probability problem is tagged Combinations and Permutations, Conditional Probability. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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