Question

If A and B are any events, the property that is not always true is: a) 0 β€ π(π΄ β© π΅) β€ 1 b) π(Ξ©) = 1 c) π(π΅) = 1 β π(π΅π) d) π(β ) = 0 e) π(π΄ βͺ π΅) = π(π΄) + π(π΅)

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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 β€ π(π΄ β© π΅) β€ 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) π(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) π(π΅) = 1 β π(π΅π)

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

d) π(β ) = 0

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

e) π(π΄ βͺ π΅) = π(π΄) + π(π΅)

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 (π(π΄ β© π΅)) from the sum of their probabilities.

Therefore, the property that is NOT always true is e) π(π΄ βͺ π΅) = π(π΄) + π(π΅).

Answer: e) π(π΄ βͺ π΅) = π(π΄) + π(π΅)

a) 0 β€ π(π΄ β© π΅) β€ 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) π(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) π(π΅) = 1 β π(π΅π)

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

d) π(β ) = 0

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

e) π(π΄ βͺ π΅) = π(π΄) + π(π΅)

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 (π(π΄ β© π΅)) from the sum of their probabilities.

Therefore, the property that is NOT always true is e) π(π΄ βͺ π΅) = π(π΄) + π(π΅).

Answer: e) π(π΄ βͺ π΅) = π(π΄) + π(π΅)

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