Two randomly selected grocery store patrons are each asked to take a blind taste test and to the state which of three diet colas he or she prefers.

Note that there are nine possible selections and so the sample space can be listed as 

 A tree diagram can be drawn for this space, representing all the possible selections as shown below. 

Both patrons prefer diet cola A = 1/3 * 1/3 = 1/9

That the two prefer the same diet of Cola, the probability space for this is represented by; AA or BB or CC

Hence the probability is 1/9 or 1/9 or 1/9 = 3/9 ~ 1/3

Prefer difference cola diets, this probability can be obtained by subtracting the probability of both persons prefering the same diet from one. 

Probability that both groceries prefer different cola diets;-  1 - 1/3  = 2/3

Alternatively, this probability can be obtained by summing the above probabilities from the tree diagram, the probabilities represented by,-

AB, AC, BA, BA, CA,CB

Which means, 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 = 6/9 = 2/3

That diet A is prefered by at least one of the patrons, the probability space is composed by the events that either one or both patrons select cola A. 

AA, or AB, or AC, or BA, or CA , probability = 1/9 or 1/9 or 1/9 or 1/9 or 1/9 = 5/9

That neither of the patrons select Cola C, should be AA, AB, BA, BB,