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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Answer to a math 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)
Hank
4.877 Answers
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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