Questions: Given Pr(A ∩ B)=0.12, Pr(A)=0.39, and Pr(B)=0.57, find Pr(A′ ∪ B).

Solution Steps

To find \(\operatorname{Pr}\left(A^{\prime} \cup B\right)\), we can use the complement rule and the formula for the union of two events. First, we find \(\operatorname{Pr}(A')\) using \(\operatorname{Pr}(A') = 1 - \operatorname{Pr}(A)\). Then, we use the formula for the union of two events: \(\operatorname{Pr}(A' \cup B) = \operatorname{Pr}(A') + \operatorname{Pr}(B) - \operatorname{Pr}(A' \cap B)\). Since \(\operatorname{Pr}(A' \cap B) = \operatorname{Pr}(B) - \operatorname{Pr}(A \cap B)\), we can substitute this into the union formula to find the desired probability.

To find \( \operatorname{Pr}(A') \), we use the complement rule: \[ \operatorname{Pr}(A') = 1 - \operatorname{Pr}(A) = 1 - 0.39 = 0.61 \]

Next, we calculate \( \operatorname{Pr}(A' \cap B) \) using the formula: \[ \operatorname{Pr}(A' \cap B) = \operatorname{Pr}(B) - \operatorname{Pr}(A \cap B) = 0.57 - 0.12 = 0.45 \]

Now, we can find \( \operatorname{Pr}(A' \cup B) \) using the union formula: \[ \operatorname{Pr}(A' \cup B) = \operatorname{Pr}(A') + \operatorname{Pr}(B) - \operatorname{Pr}(A' \cap B) = 0.61 + 0.57 - 0.45 = 0.73 \]

Final Answer