Chapter 8.3, Problem 84E

Personnel selection. To transfer into a particular technical department, a company requires an employee to pass a screening test. A maximum of $3$ attempts are allowed at $6$ -month intervals between trials. From past records it is found that $40\%$ pass on the first trial; of those that fail the first trial and take the test a second time, $60\%$ pass; and of those that fail on the second trial and take the test a third time, $20\%$ pass. For an employee wishing to transfer:

(A) What is the probability of passing the test on the first or second try?

(B) What is the probability of failing on the first 2 trials and passing on the third?

(C) What is the probability of failing on all $3$ attempts?