Find the Probability and prove that the given two events are mutually exclusive and independent or not.
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:
Group

Prefer new procedure

Prefer old procedure

No preference

Row total

Nurses

100

80

20

200

Doctors

50

120

30

200

Column total

150

200

50

400

Use the following notation for the various events: N = Nurse, D = Doctor, PN = prefer new procedure, PO = prefer old procedure, NP = no preference. Find out the following probabilities for a person selected at random from this sample of 400 people.
1. P(N), P(D), P(PN)
2. P(N, given PN), P(D, given PN)
3. P(PO, given N), P(PO, given D)
4. P(N and PN), P(D, and NP)
5. P(PN or NP)
6. Are the events PN = prefer new procedure and NP = no preference mutually exclusive? Why or why not?
7. Are the events PN = prefer new procedure and NP = no preference complementary events? Why or why not?
8. Are the events N = nurse and PN = prefer new procedure independent? Why or why not? Validate using the formula that is used to show if two events are independent or not.