In a group of 200 students, 120 like Mathematics (event M), 90 like Physics (event P) and 70 like both Mathematics and Physics.
a. (i) Construct a contingency table showing the number of students in each region of M∪P and hence determine P(M), P(P) and P(M∩P).
b. (ii) Test whether the events M and P are independent. Show all calculations clearly.
c. (iii) If the events M and P were mutually exclusive, what would P(M∩P) be? Using the given data, explain why M and P cannot be mutually exclusive.
d. (iv) Prove algebraically that two events with positive probabilities cannot be both independent and mutually exclusive.