Question

If A and B are any events, the property that is not always true is: a) 0 ≤ 𝑃(𝐴 ∩ 𝐵) ≤ 1 b) 𝑃(Ω) = 1 c) 𝑃(𝐵) = 1 − 𝑃(𝐵𝑐) d) 𝑃(∅) = 0 e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

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Hank

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To determine the property that is not always true, let's analyze each option:

a) 0 ≤ 𝑃(𝐴 ∩ 𝐵) ≤ 1

This property is always true because the probability of an intersection of two events can range from 0 (if the events are mutually exclusive) to 1 (if the events are identical).

b) 𝑃(Ω) = 1

This property is always true because the probability of the sample space, Ω, which represents all possible outcomes, is always equal to 1.

c) 𝑃(𝐵) = 1 − 𝑃(𝐵𝑐)

This property is always true because the probability of an event and the probability of its complement add up to 1.

d) 𝑃(∅) = 0

This property is always true because the probability of an empty set, represented by ∅, is always equal to 0.

e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

This property is not always true. It holds true only if the events A and B are mutually exclusive. If the events are not mutually exclusive, then we need to subtract the probability of their intersection (𝑃(𝐴 ∩ 𝐵)) from the sum of their probabilities.

Therefore, the property that is NOT always true is e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵).

Answer: e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

a) 0 ≤ 𝑃(𝐴 ∩ 𝐵) ≤ 1

This property is always true because the probability of an intersection of two events can range from 0 (if the events are mutually exclusive) to 1 (if the events are identical).

b) 𝑃(Ω) = 1

This property is always true because the probability of the sample space, Ω, which represents all possible outcomes, is always equal to 1.

c) 𝑃(𝐵) = 1 − 𝑃(𝐵𝑐)

This property is always true because the probability of an event and the probability of its complement add up to 1.

d) 𝑃(∅) = 0

This property is always true because the probability of an empty set, represented by ∅, is always equal to 0.

e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

This property is not always true. It holds true only if the events A and B are mutually exclusive. If the events are not mutually exclusive, then we need to subtract the probability of their intersection (𝑃(𝐴 ∩ 𝐵)) from the sum of their probabilities.

Therefore, the property that is NOT always true is e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵).

Answer: e) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

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