Question

Learning Objective Assessment: C6 (version 2) MATH2603: Discrete Mathematics C6: I can compute conditional probabilities, and probabilities for unions, comple- ments, or independent events. Two fair dice are thrown, one blue and one red. The events E, F, and G are defined as follows E: The sum on the two dice is less than or equal to 4. F: The sum on the two dice is even. G: Both dice come up 4. H: The red die comes up 1. Answer each question in the space provided below, you do not need to justify your answers. 1. Calculate p(E) 2. Calculate p(F). 3. Calculate p(G). 4. Calculate p(H). 5. What is p(E|H)? 6. What is p(F|H)? 7. What is p(FE)? 8. What is p(HG)? 9. Which pairs of events among E, F, and G are independent? (Note that H is not present in the list) 10. Are G and H independent?

Answer 1

- . E: The sum on the two dice is less than or equal to y. F: The sum on the two dice is even. 67: Both dice comes up 4. H: The red dice comes up E: Sum on two dice can be 2,3 or 4. (1,1), (2,1), (1, 2), (2,2), (1,3),(3,1) 6 1. Þ(E) = 36 6 F: even a 2, 4, 6, 8, 10 or la : (1.1) (1,3) (3, 1) (2,4), (1,5) (5,1) (84) 14, 2) (3, 3) (2016) (6,2) (3,5) (5,3) (4,4) (4,6) (6,4) (5,5) (6,6 ㅗ .! Þ(F) = 36 2

. 36 6 G: Both dice comes up 4 (4,4) p (G) = 36 H: The red dice comes upl. (1, 2) (1, 1) (13) (1,4) (1,5) (16) PCH) = 6 = ♡ þ (Elm) PLENH) Þ(H) PIENH) = 3 { (1, 1) (1, 2) ( (1,3)} 80. 36 .: PCE/H) = 3/6 - 3 - 1 6/36 (vi) 5 (F/H) - Þ (ENH) D(H) Þ(FNH) = 3 $(11 (1,3) (15) do 36 - P (F/H) = 3/6 =) 3 6. 2 Nii) ÞOFE) = Þ(Ens) PLE 1) (ENE) = q { (1.1) (1,3) (3,1) (2,2)} 36 P (FE) = 4/36 = 4 = 16/36 Vill) $ (H/6)= P(HOG) D(6) 3 6

þ (HOG) = 0 .: Þ(H/61) = 0 (ix) To show that two events A and B are independent we have to show that PCANB)= P(A), D(B) (a) To check E and Fare independenti PCEAF) P (F)= 18 4 36 PCE = 6 36 36 - - PCE). PCF)= - x 2 12 1 12 § PIENE) Z PCE). P(F) 9 36 2 T = 72 So E and F are not independent. (b) To check f and G are independent: P(ING) = 1 PCF) = 1 P(G)=) 36 P(FAG) I Z - PCF). PCG) 36 So f and G are also not independent. (C) To check E and G are independent P(ENG)=0, PLE)= 1 P(G) = 1 36 PLENG) 7 PCE). P (G) So E and G are not independent. (x) To check Gi and Hare independenti P(GNH) = 0 P (6) I P(4) = 1 9 36 P(GI NH) 7 P(G), P (H) So G and H are