Show that if E and F are independent so are E ESolutionIf

Show that if E and F are independent so are ? \\ E = E

Solution

If E and F are independent events,
P(EintersectionF) = P(E)*P(F)
Now P(E\'intersectionF\')
= P(EunionF)\'
= 1 - P(EunionF)
= 1 - P(E) - P(F) + P(EintersectionF)
= 1 - P(E) - P(F) + P(E)*P(F)
= [1 - P(E)] - P(F) [1 - P(E)]
= P(E\') - P(F)*P(E\')
= P(E\') [ 1 - P(F) ]
= P(E\')*P(F\')
Hence, E\' and F\' are independent.