Consider any events A, B, and C . Prove each of the...

Consider any events $A, B,$ and $C .$ Prove each of the following:
a) $P(A \cap B) \geqslant P(A)+P(B)-1$
b) $P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A \cap B)-P(A \cap C)-P(B \cap C)$
$+P(A \cap B \cap C)$

Key Concepts

Inclusion-Exclusion Principle

This principle is used to correctly compute the probability of the union of multiple events by compensating for over-counting the intersections of those events. For two events, it shows that the probability of the intersection is at least the sum of the probabilities of the events minus 1, and for three events it provides the exact adjustment needed by subtracting the probabilities of pairwise intersections and then adding back the probability of the triple intersection.

Probability Axioms

These axioms are the foundation of probability theory and state that the probability of any event is between 0 and 1, the probability of the entire sample space is 1, and that if events are mutually exclusive, the probability of their union is the sum of their individual probabilities. These principles are used to establish bounds on probabilities and guide the manipulation of expressions involving probabilities of events.

Set Theory Operations

Understanding the operations of intersection and union of sets is fundamental to probability as events are treated as sets. The concept of an intersection captures the occurrence of multiple events simultaneously while the union represents the occurrence of at least one event. These operations underpin the formulas and inequalities used in calculating combined probabilities, as seen in the problem.

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