Consider two events A and B. The probability of event A is \(P(A)=0.5\), the probability of event B is \(P(B)=0.4\), and the conditional probability of B given A is \(P(B|A)=0.6\).
(a) Find \(P(A\cap B)\).
(b) Compute \(P(A|B)\) using the result from part (a).
(c) Determine whether events A and B are independent. Justify your answer.
Question
Given that \(P(B|A)=0.6\) and \(P(A)=0.5\), find \(P(A\cap B)\).
Hint
Hint
Hint
Use the definition of conditional probability: \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Rearrange this formula to solve for \(P(A\cap B)\).
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Question
Using your answer from part (a), compute \(P(A|B)\) given that \(P(B)=0.4\).
Hint
Hint
Hint
Use the formula for conditional probability: \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Substitute the value of \(P(A\cap B)\) from part (a) and the given value of \(P(B)\).
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Question
Determine whether events A and B are independent. Justify your answer using probabilities calculated in the previous parts.
Hint
Hint
Hint
Events A and B are independent if and only if \(P(A\cap B) = P(A) \times P(B)\). Check if this condition holds using the given probabilities and the result from part (a). Alternatively, check if \(P(A|B) = P(A)\) or \(P(B|A) = P(B)\).
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Complete Answer
Events A and B are not independent because \(P(A\cap B) = 0.3\) and \(P(A) \times P(B) = 0.5 \times 0.4 = 0.2\). Since \(0.3
eq 0.2\), the condition for independence is not satisfied.
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