# Is P(A≤−B)= 1− P(A≤B) an correct equation?

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

• 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? – Pankaj Kumar Swain Apr 18 at 4:43
• In general, $P(A≤−B)= 1− P(A≤B)$ isn't correct – Fareed AF Apr 18 at 4:56

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.