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Is P(A≤−B)= 1− P(A≤B) an correct equation? If yes, kindly provide the derivation of the same.

As I get it, P(A≤−B)= 1− P(A>−B) i.e. 1−P(−A

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  • $\begingroup$ Is P(A≤−B)= 1− P(A≤B) an correct equation? If yes, kindly provide the derivation of the same. As I get it, P(A≤−B)= 1− P(A>−B) i.e. 1−P(−A<B). So, its getting a bit confusing. Please explain in details. I encountered this while going through a problem of hypothesis testing where P(Z≤−0.375) has been mentioned to equal 1−P(Z≤0.375). Is it correct? $\endgroup$ – Pankaj Kumar Swain Apr 18 at 4:43
  • $\begingroup$ In general, $P(A≤−B)= 1− P(A≤B)$ isn't correct $\endgroup$ – Fareed AF Apr 18 at 4:56
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If $B$ is positive and $A =2B$ then $P(A \leq -B)=0$ and $ 1-P(A \leq B)=1-0=1$. The event $(A \leq -B)$ is not the complement of the event $(A \leq B)$. If $E$ and $F$ are complementary events then $P(E)=1-P(F)$ but we don't have complementary events here.

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