Page-tartar E. A lumber grading system is known to be defective 243% of the time. Normally, E'it. ofthe lumber that passes through is rejected, but when it is defective, 59% ofthe pieces are rejected- In either case, pieces that are accepted are graded in the following manner: - 2% #1 Grade; Tilt-Ht? Grade; and 1M5: #3 Grade. a} Draw a tree diagram outlining each possible outcome. Please include aii of the relevant probabilities as well as the probability for each outcome. b} What is the probability of accepting a piece of lumber? c] What is the probability of getting a #2 Grade piece? d] What is the probability of accepting a piece of lumber, it it is known that the machine is not defective? e} What is the probability that a piece that was rejected came from a defective machine? Aworiring machine? Maire a brief statement about the source of rejected pieces. f} Are piece rejection rates and machine defect rates independent? Please prove mathematically