To win a basketball game, two competitors play three rounds of one three-point shot each. The series ends if one of them scores in a round, but the other misses his shot or if both get the same score in each of the three rounds. Assume competitors A and B have \(30 \%\) and \(20 \%\) of successful attempts, respectively, in three-point shots, and that the outcomes of the shots are independent events.

a. Verify the probability that the series ends in the second round is \(23.56 \%\). (Hint: Sketch a tree diagram, and write out the sample space of all possible sequences of wins and losses in the three rounds of the series, find the probability for each sequence and then add up those for which the series ends within the second round).

b. Find the probability distribution of \(X=\) number of rounds played to end the series.

c. Find the expected number of rounds to be played in the series.