Question

Brantford and Hamilton (city 1 and 2) are connected by Hwy 403 and Hamilton and Niagara Falls
(city 2 and 3) are connected by the QEW highway. Let A1, B1 denote the lanes going East and
let A2, B2 denote the lanes going West. Suppose the probability that a lane on the 403 will not
need any major repair in the next year is 80%. For the QEW, the probability is 90%.

Brantford and Hamilton (city 1 and 2) are connected by Hwy 403 and Hamilton and Niagara Falls (city 2 and 3) are connected by the QEW highway. Let A1, B1 denote the lanes going East and let A2, B2 denote the lanes going West. Suppose the probability that a lane on the 403 will not need any major repair in the next year is 80%. For the QEW, the probability is 90%. 2 403 (a) What is the probability that the 403 wil need major repair in the next year? Assume that if one lane needs repair, the probability the other lane will also need repair is 3 times the original probability. [3 marks] (b) What is the probability that the QEW will need major repair in the next year? Assume that if one lane needs repair, the probability the other lane will also need repair is 3 times the original probability. [3 marks] (c) If the need for repair on the 403 and QEW are independent, what is the probability that the route between Brantford and Niagara Falls will need major repair in the next year? 2 marks]

Answer 1

a) Observe that for highway it is given that if one lane needs repair then the other lane needs repair with probability 3 x 0.2 0.6. Hence the probability that both lane needs repair is 0.6 x 0.2 0.12 Probability of highway needing repair is (0.2 + 0.2-0.12-0.28. b) Siilarly the probability of highway QEW needing major repair is (0.1 + If the need for repair on those two highways are independent then the probability of getting the road from Brantfort and Niagra Falls will need major repair next year will be (0.28 + 0.17-0.28 x 0.17) 0.4024