The buses are sometimes crowded and sometimes noisy,both of which are problems for Tom as he likes to use the bus journeys to do his Stats assignments.When the bus is on time,it is crowded with probability 0.5.When it is late,it is crowded with probability 0.7.The bus is noisy with probability 0.8 when it is crowded,and with probability 0.4 when it is not crowded. Question(b)Formulate events C and N corresponding to the bus being crowded and noisy.Do the events C and N form a partition of the sample space?Explain why or why not. Solution.LetC=“crowded,N=“noisy' C and N do NOT form a partition of It is possible for the bus to be noisy when it is crowded,so there must be some overlap between C and N. Question(c)Write down probability statements corresponding to the information given above.Your answer should involve two statements linking C with T and L,and two statements linking N with C. Solution. P(C1T)=0.5: PC1L)=0.7. P(N1C)=0.8:P(N1C=0.4. Questin(d)Find the probability that the bus is crowded. P(C)=P(C|T)P(T)+P(C L)P(L)(Partition Theorem) =0.5×0.6+0.7×0.4 =0.58. The buses are sometimes crowded and sometimes noisy, both of which are problems for Tom as he likes to use the bus journeys to do his Stats assignments. When the bus is on time, it is crowded with probability 0.5. When it is late, it is crowded with probability 0.7. The bus is noisy with probability 0.8 when it is crowded, and with probability 0.4 when it is not crowded. Question(b) Formulate events C and N corresponding to the bus being crowded and noisy. Do the events C and N form a partition of the sample space? Explain why or why not. Solution. Let C = “crowded”, N =“noisy”. C and N do NOT form a partition of  . It is possible for the bus to be noisy when it is crowded, so there must be some overlap between C and N. Question(c) Write down probability statements corresponding to the information given above. Your answer should involve two statements linking C with T and L, and two statements linking N with C. Solution. Questin(d) Find the probability that the bus is crowded