A judicial case can be heard at three levels: lower court (1), appellate court (2), and high court (3). State 4 in the transition diagram below represents final termination of the case. The weights in the directed graph are the probabilities that the case will move from one court to another, e.g., an appellate court case will return there with probability \(1 / 4\), or be appealed to high court with probability \(3 / 4\). Find the probability that a case that begins in lower court is finally terminated after exactly \(k\) hearings.

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3 2/3 1/4 3/4 1/3 Exercise 6