Identify the probability statements that would allow us to conclude the events are independent. Check all that apply.
P(A|BC) = P(A)
P(B|A) = P(A|B)
P(B|A) = P(B)
P(A|B) = P(A|BC)
P(A|B) = P(B)
P(A|B) = P(A)



answer is a c d f

The probability statements that would allow us to conclude that the events are independent are P(B|A) = P(B) and P(A|B) = P(A).

To determine if two events are independent, we need to check if the probability of one event is affected by the occurrence of the other event. If the probability of one event remains the same, regardless of whether the other event occurs or not, then the events are independent.

Let's analyze each of the given probability statements and see which ones would allow us to conclude that the events are independent.

P(A|BC) = P(A):

This statement indicates the probability of event A occurring given that both events B and C have occurred.

We cannot conclude independence from this statement, as the occurrence of events B and C may affect the probability of A.

P(B|A) = P(A|B):

This statement indicates the probability of event B occurring given that event A has occurred, is equal to the probability of event A occurring given that event B has occurred.

This is the definition of conditional probability, and it does not provide enough information to determine the independence of the events.

P(B|A) = P(B):

This statement indicates the probability of event B occurring given that event A has occurred is equal to the marginal probability of event B.

This would only be true if the occurrence of event A has no effect on the probability of event B, which would indicate independence.

P(A|B) = P(A|BC):

This statement indicates the probability of event A occurring given that event B has occurred is equal to the probability of event A occurring given that both events B and C have occurred.

This statement does not provide enough information to determine the independence of the events.

P(A|B) = P(B):

This statement indicates the probability of event A occurring given that event B has occurred is equal to the marginal probability of event B.

As previously mentioned, this would only be true if the occurrence of event A has no effect on the probability of event B, which would indicate independence.

P(A|B) = P(A):

This statement indicates the probability of event A occurring given that event B has occurred is equal to the marginal probability of event A.

This would only be true if the occurrence of event B has no effect on the probability of event A, which would indicate independence.

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