I taught my daughter to drive and she was a bit heavy on the brakes to start. During drives to and from her school, there are 16 locations requiring braking (e.g., roundabouts, stop signs, slip lanes etc.). Further, a school term has 50 days, meaning 100 total drives back and forth. Therefore 1600 braking incidents.

Assume the wear on the brake pads from each braking instance has a mean of 0.009mm and standard deviation of 0.025mm.

a) If the lining of my brake pads is 16.5mm thick at the start of a term, what is the approximate chance they last out the term (assuming my daughter misses no days of school)? [2 marks]

b) In fact, wear is uneven between front and rear pads. Suppose total wear on the rear pads during a single trip is normal with mean 0.16mm and standard deviation 0.12mm, while total wear on the front pads is normal with mean 0.128mm and standard deviation 0.08mm. Further, assume the correlation between wear on the pads is 0.8. If the rear pad was worn down by 0.192mm during today's morning drive, what is the probability the front pad wear was less than 0.16mm? [2 marks]