Answer
$P(one~is~diet~and~one~is~regular)=\frac{9}{22}\approx0.40909$
Work Step by Step
The events "both cans contain diet soda" and "both cans contain regular soda" are mutually exclusives (disjoint events).
Using the Addition Rule for Disjoint Events (page 270):
$P(both~cans~contain~diet~soda~or~both~cans~contain~regular~soda)=P(both~cans~contain~diet~soda)+P(both~cans~contain~regular~soda)=\frac{1}{22}+\frac{6}{11}=\frac{1}{22}+\frac{12}{22}=\frac{13}{22}$
The event "one is diet and one is regular" is the complement of "both cans contain diet soda or both cans contain regular soda".
Using the Complement Rule (see page 275):
$P(one~is~diet~and~one~is~regular)=1-P(both~cans~contain~diet~soda~or~both~cans~contain~regular~soda)=1-\frac{13}{22}=\frac{22}{22}-\frac{13}{22}=\frac{9}{22}\approx0.40909$
