8. Alice, Betty and Catherine take an economy test. The probability for each of them pass the test are 1, 2 and respectively. (a) Draw a tree diagram. Write out the sample space. (b) Calculate the probability that two of them fail the test. (c) Find the probability that at least two of them pass the test. 9. Items leaving a production process are tested automatically, but it is known that the test is not completely reliable. Experience shows that 85% of the output is satisfactory and 15% is defective. The probability that a satisfactory item fails the automatic test is 0.07 and the probability that a defective item fails the automatic test is 0.97. (a) Draw a well-labelled probability tree diagram with joint probabilities calculated to illustrate this situation. (b) Calculate the probability that a randomly selected item leaving the process (i) fails the test, (ii) is defective given that it fails the test, is satisfactory given that it does not fail the test, (iv) Is satisfactory or it does not fail the test