Question

Exercise 3. [ For those who want a little challenge.] If E and F are independent events, show that E and Fc are independent events. (Intuitively this makes sense: if knowing F happens doesn’t give any extra insight into whether E happens, then surely knowing F doesn’t happen doesn’t provide any useful information about the occurrence of E either!) Consider the handedness of human beings to be either either left or right. Why does it make sense to simply ask: Are sex and handedness independent?, as opposed to asking about the independence of all four combinations?

Answer 1

$E~and~F~are~independent~iff~P(E\cap F)=P(E)P(F).\\ Now,~P(E)=P((E\cap F)\cup(E\cap F^c))\\ =P(E\cap F)+P(E\cap F^c)~since~E\cap F~and~E\cap F^c~are~disjoint\\ or,~P(E\cap F^c)=P(E)-P(E\cap F)\\ =P(E)-P(E)P(F),~since~E~and~F~are~independent\\ =P(E)(1-P(F))=P(E)P(F^c)\\ Hence ~E~and~F^c~are~also~independent.$

$4~combinations~are:~(Male,~left),~(Male,~Right),\\ (Female,~left),~(Female,~Right).\\ If~P(Male,left)=P(Male)P(left),~P(Female,left)=P(Male)P(left),\\ P(Male,Right)=P(Male)P(Right),~P(Female,Right)=P(Female)P(Right)\\ then~we~say~that~sex~does~not~influence~handedness~i.e.~they~are~independent.\\ Hence~it~is~reasonable~to~ask~:are~sex~and~handedness~independent?$