For the following claims, state whether they are TRUE or FALSE. Statements claimed to be TRUE must be accompanied by a proof, and statements claimed to be FALSE must be accompanied by a counterexample.

(a) Let A and B be events in the sample space ?. Then P(A ? B) ? P(A)P(B). [2 marks]

(b) LetEbeaneventwith0

sample space S. Then PE satisfies the axioms of probability. [2 marks]

(c) Let A and B be events in the sample space ?. Then P(A?BC) = P(A?B)?P(B). [3

marks]

(d) Let A1,A2,...,An be events in the sample space ?. Then P (?ni=1Ai) ? min{P (Ai)}.

(When answering the questions, writing down the final answer will not be sufficient to receive full marks. Please show all calculations unless otherwise specified. Also define any events and notation that you use in your solutions.)

For the following claims, state whether they are TRUE or FALSE. Statements claimed to be TRUE must be accompanied by a proof, and statements claimed to be FALSE must be accompanied by a counterexample. (a) Let A and B be events in the sample space . Then P(An B)