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.
390 exam-style questions on OCR (MEI) A Level Maths 2.4 Probability Distributions, covering 2.4.1 Recognise binomial situations, 2.4.2 Probability of success p, 2.4.3 Calculate binomial probabilities, 2.4.4 Mean of the binomial distribution, 2.4.5 Expected frequencies for binomial, 2.4.6 Probability functions and discrete random variables, 2.4.7 Numerical probabilities for a simple distribution, 2.4.8 Normal distribution as a model (A-level only), 2.4.9 Shape of the Normal curve (A-level only), 2.4.10 Linear transformation and standardising (A-level only), 2.4.11 Symmetry and inflection of Normal curve (A-level only), 2.4.12 Calculate probabilities from a Normal distribution (A-level only), 2.4.13 Model with probability distributions, and 2.4 Probability Distributions. Each one has a worked solution and a mark scheme showing where the marks go.