Check whether the following probabilities $P(A) = 0.5$,$P(B) = 0.4$,and $P(A \cup B) = 0.8$ are consistently defined.
(A) Given: $P(A) = 0.5$,$P(B) = 0.4$,and $P(A \cup B) = 0.8$.
For any two events $A$ and $B$,the probability of their union is given by the formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
Substituting the given values: $0.8 = 0.5 + 0.4 - P(A \cap B)$.
$0.8 = 0.9 - P(A \cap B) \implies P(A \cap B) = 0.9 - 0.8 = 0.1$.
Since $0 \leq P(A \cap B) \leq P(A)$ and $0 \leq P(A \cap B) \leq P(B)$,and here $0.1 \leq 0.5$ and $0.1 \leq 0.4$,the values are consistent.
For any two events $A$ and $B$,the probability of their union is given by the formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
Substituting the given values: $0.8 = 0.5 + 0.4 - P(A \cap B)$.
$0.8 = 0.9 - P(A \cap B) \implies P(A \cap B) = 0.9 - 0.8 = 0.1$.
Since $0 \leq P(A \cap B) \leq P(A)$ and $0 \leq P(A \cap B) \leq P(B)$,and here $0.1 \leq 0.5$ and $0.1 \leq 0.4$,the values are consistent.
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