In a particular manufacturing plant, two machines (A and B) produce a particular part. One machine (B) is newer and faster. In one five-minute period, a lot consisting of 32 parts is produced. Twenty-two are produced by machine B and the rest by machine A. Suppose an inspector randomly samples a dozen of the parts from this lot.

a. What is the probability that exactly three parts were produced by machine A?

b. What is the probability that half of the parts were produced by each machine?

c. What is the probability that all of the parts were produced by machine B?

d. What is the probability that seven, eight, or nine parts were produced by machine B?