for which of the following probability assignments are events a and b independent

 To determine whether events A and B are independent, you need to check whether the probability of the occurrence of one event is affected by the occurrence or non-occurrence of the other event. Mathematically, two events (A and B) are considered independent if:

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P(A∩B)=P(A)⋅P(B)

In words, the probability of both events A and B occurring is equal to the product of the probabilities of each event occurring independently.

If this equation holds true, then events A and B are independent. If not, they are dependent.

It's worth noting that another way to express independence is:

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P(A∣B)=P(A)

In words, the probability of event A occurring given that event B has occurred is equal to the probability of event A occurring independently.

Without specific information about the probability assignments for events A and B, it's challenging to provide a definitive answer. If you have the specific probabilities for each event and the joint probability, you can apply the formulas above to check for independence.