Refer to the sleep apnea study published in Chance (Winter 2009), Exercise. Recall that the various stages of sleep for a large group of sleep apnea patients were monitored in 30-second €œepochs,€ and the sleep stage for the previous and current epoch was determined. A summary table is reproduced below. One of the epochs in the study is selected.

Given that the previous sleep stage for the epoch was the Wake state, what is the probability that the current sleep stage is REM?
b. Given that the current sleep stage for the epoch is REM, what is the probability that the previous sleep stage was not the Wake state?
c. Are the events {previous stage is REM} and {current stage is REM} mutually exclusive?
d. Are the events {previous stage is REM} and {current stage is REM} independent?
e. Are the events {previous stage is Wake} and {current stage is Wake} independent?

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Previous Sleep Stage Current Stage Non-REM REM Wake Non-REM REM Wake 31,880 160 1,733 346 7,609 175 1,588 358 6,079 Totals 33,814 8,127 7,987