# Events A and B are independent, events A and C are mutually exclusive, and events B and C are independent. If P(A) = 1/2; P(B) = 1/4; P(C) = 1/8; what is P(A∪B∪C)?

Events $A$ and $B$ are independent, events $A$ and $C$ are mutually exclusive, and events $B$ and $C$ are independent.
If ; what is $P\left(A\cup B\cup C\right)$?
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Franklin Frey
Note that the event $A\cup B\cup C$ can happen in the following two disjoint ways: (i) $B$ holds or (ii) $B$ fails and one of $A$ or $C$ holds.
The probability of (i) is $1/4$.
For probability of (ii) we need to work somewhat harder. Since $A$ and $C$ are mutually exclusive, we want $Pr\left({B}^{\prime }\cap A\right)+Pr\left({B}^{\prime }\cap C\right)$. Here ${B}^{\prime }$ denotes the complement of $B$. The required probabilities can be found by independence.
It remains to put the pieces together.