Question: (16 pts) A new postoperative procedure is administered to a number of patients in a large hospital. A researcher selected nurses and doctors and asked which procedure each person preferred. The results are:

Use the following notation for the various events: N = Nurse, D = Doctor, PN = prefer new procedure, PO = prefer old procedure, NP = no preference. Find the following probabilities for a person selected at random from this sample of 400 people.

a.) P(N)

b.) P(D)

c.) P(PN)

d.) P(N | PN)

e.) P(D | PN)

f.) P(PO | N)

g.) P(PO | D)

h.) P(N and PN)

i.) P(D and NP)

j.) P(PN or NP)

k.) Are the events PN = prefer new procedure and NP = no preference mutually exclusive? Why or why not?

l.) Are the events PN = prefer new procedure and NP = no preference complementary events? Why or why not?

m.) Are the events N = nurse and PN = prefer new procedure independent? Why or why not? Justify using the formula that is used to prove if two events are independent or not.