1 Probability calculations {a} A rm that services private vehicles has noticed that 24.9% of vehicles require tyre changes when they come in for service, while 15.3% need repairs to the wipers andf or washer system and 72.1% require neither. i] [2 marl-is} Find the probability that a randomly selected vehicle requires both a tyre change and wiper;I washer repairs when it comes in for service. ii) [1 mark] Hence nd the probability that a vehicle which needs a tyre change also needs wiper,\" washer repairs. iii} {1 mark) Based on your answers above. are the events \"needs tyre change\" and \"needs wiper} washer repairs'1 independent? Provide a brief explanation of your conclusion. {b} i} [2 marks} The time to failure T of an electrical component has a distribution with the following probability density function (pdf): r) = hr\" fort :3 2 Find the value of it: which makes this a valid pdf. ii) [3 marks} The number of customers visiting a specialty store each hour is denoted N and has the following probability distribution: 'n. 5 7 S 9 Use this information and the fact that ENJ = 6.4 to determine the variance Var N} of this variable. {c} [3 marks} The service times in a store are observed to follow an exponential distribution with an average service rate of A = 1.25 customers per minute. Find the probability that the service time for a randomly chosen customer is between [1.5 and 2.5 minutes