Event A and B are independent of each other...
What does independent mean here? How would it change if it said dependent. The full question continues as:
The probability of event A occurring is 3x as likely as event B occurring. The probability that neither event A nor event B will occur is 7/27. What is the probability of event A occurring?

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If two events A and B are independant then
P(A intersection B)=P(A)P(B)

Just take P(B) as x P(A)=3x.
Apply the condition and solve for x.

phi the venn diagram i think is wrong, independant does not mean that it has no element in common.

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