Determine event independence using P(A and B)=P(A)P(B).
Determine whether two events are independent using \(P(A and B) = P(A)P(B)\)
Problem
Are events \(A\) and \(B\) independent? Use these probabilities: \(P(A)=0.4\), \(P(B)=0.5\), \(P(A~\text{and}~B)=0.3\).
Big Picture
What this problem is really about
Independence requires the given joint probability to equal the product of the two marginal probabilities. We’ll compute that product separately, compare it with the reported probability of simultaneous occurrence, and classify the events from exact equality or mismatch. Keeping these two quantities distinct prevents the joint value from being assumed rather than tested.
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