Probability of the complement

Proving the complement rule from probability axioms

This easy question tests whether you can work backwards from first principles in probability theory. Rather than memorizing formulas, it asks you to derive a fundamental identity using only the axioms—a skill that reveals whether you truly understand the foundations rather than just the tools.

To solve it, you'll need to recall the three axioms of probability, recognize how the sample space and complement partition it, and apply the axioms systematically to derive the relationship. The second part follows naturally once you've completed the derivation, requiring you to consider a degenerate case and what it means for the probability measure.

• Kolmogorov's axioms and their implications
• Partition and additivity
• The complement and the sample space