Safety checks on the roads outside the university are conducted on a daily basis. It is...

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Safety checks on the roads outside the university are conducted on a daily basis. It is known that on average, 0.5 potholes are found per day on the roads. (a) (b) Find the probability that on a randomly chosen day, at least 2 potholes are found. (4 marks) Find the probability that in a randomly chosen period of 20 days, there are exactly 7 days in which at least 2 potholes are found. (5 marks) Cracks on the road are also found an average rate of 0.75 per day. (c) Find the probability that in a randomly chosen period of 20 days, there is at most 2 potholes found given that a total of 10 faults (cracks + potholes) are discovered. State any assumptions you may have made. (8 marks) When a pothole or a crack is discovered, there is 0.4 chance that it will be repaired by the end of the same day. (d) Find the probability on a randomly chosen day, the holes/cracks discovered will not be able to all be repaired by the end of the day. (8 marks) Remark: For Question 5(d), you may use the following identity: Σ =1+x+ k! k=0 2! 3! +... = e*. Safety checks on the roads outside the university are conducted on a daily basis. It is known that on average, 0.5 potholes are found per day on the roads. (a) (b) Find the probability that on a randomly chosen day, at least 2 potholes are found. (4 marks) Find the probability that in a randomly chosen period of 20 days, there are exactly 7 days in which at least 2 potholes are found. (5 marks) Cracks on the road are also found an average rate of 0.75 per day. (c) Find the probability that in a randomly chosen period of 20 days, there is at most 2 potholes found given that a total of 10 faults (cracks + potholes) are discovered. State any assumptions you may have made. (8 marks) When a pothole or a crack is discovered, there is 0.4 chance that it will be repaired by the end of the same day. (d) Find the probability on a randomly chosen day, the holes/cracks discovered will not be able to all be repaired by the end of the day. (8 marks) Remark: For Question 5(d), you may use the following identity: Σ =1+x+ k! k=0 2! 3! +... = e*.