Question 2. a) A certain delivery service offers both express and standard delivery. Seventy-five percent of parcels are sent by standard delivery, and 25% are sent by express or those sent standard, 80% arrive the next day, and of those sent express, 95% arrive the next day. A record of a parcel delivery is chosen at random from the company's files. i) What is the probability that the parcel was shipped express and arrived the next day? [4 marks] ii) What is the probability that it arrived the next day? [4 marks] iii) Given that the package arrived the next day, what is the probability that it was sent express? [4 marks) PTO 3 b) Batches that consist of 50 coil springs from a production process are checked for conformance to customer requirements. The probability of a spring being non- conformant is 0.10. Assume that the number of nonconforming springs in a batch, denoted as X, is a binomial random variable. Use the relevant output below to calculate the P(X249). [3 marks) dbi nom(x=49, si ze=50, prob=0.90) [1] 0. 02863208 >dbi nom(x=49, si ze=50, prob=0. 10) [1] 4. 5e-48 > dbi nom(x=50, si ze=50, prob=0. 10) [1] le-50 sum(dbi nom( x=49:50, si ze=50, prob=0.90)) [1] 0.03378586 c) The number of cracks in a section of a particular motorway that need repair is assumed to follow a Poisson distribution with a mean of 5 cracks per km. i) Use the relevant output below to calculate the probability that there are 10 cracks that require repair in 2kms. [3 marks) dpoi s(x=10 lambda=5) [1] 0.01813279 > dpoi s(x=5 lambda=5) [1] 0.1754674 > dpoi s(x-10, lambda-10) [1] 0. 12511 ii) If the number of cracks is related to vehicle load on the motorway and some sections of the motorway have a high load of vehicles whereas other sections carry a light load, how justified is the assumption of a Poisson distribution for the number of cracks that need repair? [2 marks]