A quality characteristic is monitored by a control chart designed so the probability that a certain out-of-control condition will be detected on the first sample following the shift to that state is 1-beta. Find the following: a) The probability that the out-of-control condition will be detected on the second sample following the shift b) The probability that the out-of-control condition will be detected on the mth sample following the shift c) The expected number of subgroups analyzed before the shift is detected. d) The probability that the first sample following the shift produces a statistic that plots inside the control limits. e) The probability that the shift is not detected on the mth subsequent sample