1. A sample space has four elements. For the potential probability functions for the sample space below, state whether they are valid or not. Provide support for your response.

(a) P(1) = 0.6, P(2) = 0.05, P(3) = 0.4, P(4) = 0.2.

(b) P(1) = 0.5, P(2) = 0.2, P(3) = 0.2, P(4) = 0.1.

(c) P(1) = 0.15, P(2) = 0.3, P(3) = 0.1, P(4) = 0.45.

(d) P(1) = 0.3, P(2) = 0.3, P(3) = 0.2, P(4) = 0.6.

2.For three events A, B, and C, the following is known: P(A) = P(B) = P(C), P(BC) = 0.2P(B C), P(AB) = P(AC) = 0, and the probability of no events occurring is 0.1

(a) Sketch a Venn diagram that matches the information provided about the three events.

(b) Find P(B).

(c) Find P(ABC).

(d) Find P[(A B) Cc].