Question

If A and B are any events, the property that is not always true is: a) 0 ≀ 𝑃(𝐴 ∩ 𝐡) ≀ 1 b) 𝑃(Ξ©) = 1 c) 𝑃(𝐡) = 1 βˆ’ 𝑃(𝐡𝑐) d) 𝑃(βˆ…) = 0 e) 𝑃(𝐴 βˆͺ 𝐡) = 𝑃(𝐴) + 𝑃(𝐡)

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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 ∩ 𝐡) ≀ 1 b) 𝑃(Ξ©) = 1 c) 𝑃(𝐡) = 1 βˆ’ 𝑃(𝐡𝑐) d) 𝑃(βˆ…) = 0 e) 𝑃(𝐴 βˆͺ 𝐡) = 𝑃(𝐴) + 𝑃(𝐡)

Expert avatar
Hank
4.8
62 Answers
To determine the property that is not always true, let's analyze each option:

a) 0 ≀ 𝑃(𝐴 ∩ 𝐡) ≀ 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(Ξ©) = 1
This property is always true because the probability of the sample space, Ξ©, which represents all possible outcomes, is always equal to 1.

c) 𝑃(𝐡) = 1 βˆ’ 𝑃(𝐡𝑐)
This property is always true because the probability of an event and the probability of its complement add up to 1.

d) 𝑃(βˆ…) = 0
This property is always true because the probability of an empty set, represented by βˆ…, is always equal to 0.

e) 𝑃(𝐴 βˆͺ 𝐡) = 𝑃(𝐴) + 𝑃(𝐡)
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 ∩ 𝐡)) from the sum of their probabilities.

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

Answer: e) 𝑃(𝐴 βˆͺ 𝐡) = 𝑃(𝐴) + 𝑃(𝐡)

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