If C1 and C2 are subsets of the sample space 𝒞, show that P(C1 ∩C2) ≤P(C1) ≤P(C1 ∪C2) ≤P(C1)+P(C

step 1 In problem 7, C1 and C2 are two subsets from the sample space C. We want to prove the following.

If C1 and C2 are subsets of the sample space 𝒞, show that...

If $C_{1}$ and $C_{2}$ are subsets of the sample space $\mathcal{C}$, show that
$$
P\left(C_{1} \cap C_{2}\right) \leq P\left(C_{1}\right) \leq P\left(C_{1} \cup C_{2}\right) \leq P\left(C_{1}\right)+P\left(C_{2}\right)
$$

Key Concepts

Probability Measure

A probability measure is a function that assigns a non-negative real number to each event in a probability space, satisfying the axioms of non-negativity, normalization (the probability of the entire sample space is 1), and countable additivity. This concept underpins the relationship between probabilities of events and is essential for the derivations and inequalities involving unions and intersections of events.

Event Operations (Union and Intersection)

In probability theory, events are subsets of the sample space, and operations such as union and intersection are used to combine these events. The union represents the occurrence of at least one of the events, while the intersection represents the simultaneous occurrence of both events. Understanding these operations is crucial for applying probability formulas and establishing relationships among the probabilities of combined events.

Monotonicity of Probability

Monotonicity is the property of a probability measure whereby if one event is a subset of another, then the probability of the first event is less than or equal to the probability of the second. This concept is key in establishing inequalities between the probabilities of intersections, individual events, and unions of events.

Subadditivity

Subadditivity refers to the property that the probability of the union of any two events is less than or equal to the sum of the probabilities of the individual events. This concept is a direct consequence of the axioms of a probability measure and is fundamental for proving the upper bound in the given inequality.

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