4.53 Independence of complements.
Show that if events and obey the multiplication rule, , then and the complement of also obey the multiplication rule, . That is, if events and are independent, then and are also independent. (Hint: Start by drawing a Venn diagram and noticing that the events “ and ” and “ and ” are disjoint.)
4.53
Because ( and ) and ( and ) are disjoint, then can be written as subtraction gives Because and are independent, applying the multiplication rule gives thus and are independent.