The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low. Historically, the prob ability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively. Suppose that 20 bolts are evaluated and that the torque ratings are independent.

(a) What is the probability that 12, 6, and 2 bolts are rated as high, moderate, and low, respectively?

(b) What is the marginal distribution of the number of bolts rated low?

(c) What is the expected number of bolts rated low?

(d) What is the probability that the number of bolts rated low is greater than two?

(e) What is the conditional distribution of the number of bolts rated low given that 16 bolts are rated high?

(f) What is the conditional expected number of bolts rated low given that 16 bolts are rated high? (g) Are the numbers of bolts rated high and low independent random variables?