An electronics manufacturer is monitoring the production of specialized micro-drills. Each batch contains 50 drills, and a drill is classified as defective if it fails a precision test. A quality control inspector takes a random sample of 200 batches and records the number of defective drills, xxx, in each batch. The results are summarized in the table below:
| Number of defective drills (xxx) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ≥10\ge 10≥10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Frequency (fff) | 2 | 10 | 20 | 35 | 45 | 38 | 25 | 15 | 7 | 3 | 0 |
Calculate the mean and the variance of these data.
Explain why the results in part (a) suggest that a Binomial distribution may be a suitable model for the number of defective drills per batch.
The manufacturer uses a Binomial distribution X∼B(50,0.1)X \sim B(50, 0.1)X∼B(50,0.1) to model the number of defective drills per batch.
For a randomly selected batch find, using this model, the probability that there are (i) at most 2 defective drills, (ii) at least 3 but no more than 6 defective drills.
A large-scale production run consists of 200 drills.
Using the manufacturer's model and a suitable approximation, show that the probability that there are more than 25 defective drills in a production run is 0.10 to 2 decimal places. Show your working clearly. (Solutions relying on calculator technology are not acceptable.)
A period of 15 production runs is selected at random.
Find the probability that in this period there are exactly 3 production runs that have more than 25 defective drills. Show your working clearly.
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.