The car ends
within a half mile of the sign indicating 2.5 miles means that the car should end in
one mile interval, between the signs indicating 2 (2.5-0.5=2) and 3 miles (2+0.5=3), so within the red segment on the diagram.
Now if the car appears somewhere between the blue dots, between 1.5 and 2 miles signs then after traveling 0.5 miles the car will be in the red segment. So in order after traveling 0.5 miles the car to end within the red segment it should appear between 1.5 and 2.5 miles, so within 1 mile interval, as the circumference of the track is 3 miles then the probability of that will be P=favorable/total=1/3. As I mentioned in my previous post actually it doesn't matter where the car appears or what distance it travel, as long as favorable interval in the end is 1 mile and total interval is 3 miles then the probability will be 1/3 miles.
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