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2.4 Probability Distributions

2.4 Probability Distributions

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Question 69

An industrial chemical process consists of three sequential stages: preparation, reaction, and filtration. The durations in minutes for these stages are modeled as independent normal random variables:

P∼N(40,2.52)P \sim \text{N}(40, 2.5^2)P∼N(40,2.52) represents the preparation time, R∼N(110,6.42)R \sim \text{N}(110, 6.4^2)R∼N(110,6.42) represents the reaction time, F∼N(75,4.22)F \sim \text{N}(75, 4.2^2)F∼N(75,4.22) represents the filtration time.

a.

Determine the probability that the total time for a randomly selected sequence of all three stages exceeds 240 minutes.

[4]
b.

Calculate the probability that the filtration stage is at least 30 minutes longer than the preparation stage for a randomly selected batch.

[3]
c.

Given that P(P+R+F<t)=0.05P(P + R + F < t) = 0.05P(P+R+F<t)=0.05, find the value of ttt.

[3]
d.

In a single manufacturing shift, 6 independent batches are processed. A quality control officer uses the probability calculated in part (a) to determine the probability that the total processing time exceeds 240 minutes in at least two of these batches. Find the value the officer should obtain.

[4]
e.

An engineer, Dr. Aris, suggests that the assumption of independence between stages within the same batch is likely flawed. Explain with a reason whether the use of the answer from part (a) in the calculation for part (d) is appropriate.

[1]
Markscheme

2.4 Probability Distributions Questions

  1. A Level
  2. /Maths
  3. /2.4 Probability Distributions

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.

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