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# 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
4.8
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$𝐵$

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