Question

Learning Objective Assessment: C6 (version 3) MATH2603: Discrete Mathematics C6: I can compute conditional probabilities, and probabilities for unions, comple- ments, or independent events. Five fair coins will be flipped; each coin is a different color. You may not use a calculator, but you may also leave your answer as a sum, product, and/or quotient of integers. You do not need to simplify. The events E, F, and G are defined as follows: E: There are an even number heads, F: The total number of heads is greater than or equal to 4, G: All five coins come up heads, H: The first coin comes up heads. 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(E|F)? 8. What is p(G|H)? 9. Which pairs of events among E, F, and G are independent? (Note that G is not present in the list) 10. Are G and H independent?

Answer 1

Here five coins be slipped Them tokal number of sample (5) = 5 - 32 space Now where Bi is defined Consider B, B1, B2, B3, B4, Bs events as the event of getting exactly & heats, 5 (independent and hair) coin tossed out of Then there Sei exactly are sequence of 5 draws observing that ontaining i heads. . n(E) = n (Bj v B₂ v Bu) [ all are independent 5c 11 Seu - 2 1 + 10+5 TI = 16 16 PE) = ne n(s) 32 = 2 n (F) = seu = 5+1=6 n (Bu UB) PLE)= N(E) n(s) 6 32 16 n (B5) Ses 1 NG) - P (G) = n(s) 32 n (H) = uei ucq 4e3 ueu 11 0742 P(H)= NH) n(s) 32 Now PE/H) = Р(ЕОН) P (H) n EnH) n (14) • Again n (ENH) = 4e, uez - 4 +4 = 8 P (€/H) = 12

n (FNH) = 4e z then 4+1=5 P (F/H) - NINH m(H) 읍 16 EnF) = sen 5 p(F/F) - n (EnF) NFO info DO 1 GRH) PG/H) = ncanH NCHO He 1 16 o and G are independent oll ell 16 No, G and H are not independent,