A robotic arm is used to sort precision components in a factory. The reliability of the arm depends on the sensor distance s s\,s mm. The probability that the arm successfully places a component in a single attempt is ppp. For distances between 20 mm and 200 mm, the probability is given by p=1200(220−s)\displaystyle p = \frac{1}{200}(220 - s)p=2001(220−s). Assume each attempt is independent.
In a test run, the arm is set to a sensor distance of 140 mm and attempts to place 15 components.
(i) Find the probability that exactly 7 components are placed successfully.
(ii) Find the probability that at least 10 components are placed successfully.
In a second trial, the arm is set to a distance of x x\,x mm and attempts 20 placements.
Find, to the nearest integer, the value of x x\,x if the arm has a 99% chance of succeeding at least once.
In a final production run of 120 components, the arm is set to a distance of 115 mm.
Using a suitable approximation, estimate the probability that the arm fails to place the component at most 50 times.
