Suppose you roll a regular fair die twice. Let Si represent the event that the sum of the two rolls is i for i= 2, ..., 12 and E represent the event that their product is an odd number. That is, S; = {(w1,W2) : W1 +w2 = i,w1 E 11, W2 E 12}, and E = {(w1,W2) : (w1 X W2) is an odd number, wi EN1, wel2}, where N1 = {1, 2, 3, 4, 5, 6} = 12 representing the sample space of each roll. a) (2 points) Draw a Venn diagram representing the sample space 2 {(w1,W2) : W1 EN1,w2 E 122}. Clearly indicate the events Si, E, and their intersections, and make sure that the Venn diagram does not contain any areas for empty sets. Label the Venn diagram with N, Si for all i, and E. b) (3 points) Compute P(Si) for all i and P(E). c) (3 points) Compute P(Sin E) for all i. d) (1 points) Are S4 and E independent? Justify. e) (1 points) Is the event that the sum of the two rolls is either 10 or 12 independent of E?