A,B,C: probability exactly one of A,B occurs is (1-k), exactly one of B,C is (1-2k), exact…

Question

$A,B,C$: probability exactly one of $A,B$ occurs is $(1-k)$, exactly one of $B,C$ is $(1-2k)$, exactly one of $C,A$ is $(1-k)$, and $P(A\cap B\cap C)=k^2$, $0<k<1$. Find $P(A\cup B\cup C)$.

Accepted Answer

Adding the three "exactly one" equations and simplifying: $\sum P(A)-\sum P(A\cap B) = \dfrac{3-4k}{2}$.

$$P(A\cup B\cup C) = \frac{3-4k}{2}+k^2 = (k-1)^2+\frac12 > \frac12$$

Solutions · 1