The independent random variables MAM_AMA and MBM_BMB represent the masses, in grams, of two chemical reagents used in a laboratory process:
MA∼N(80,82)andMB∼N(35,42) M_A \sim \text{N}(80, 8^2) \quad \text{and} \quad M_B \sim \text{N}(35, 4^2) MA∼N(80,82)andMB∼N(35,42)A residual mass RRR is calculated using the formula R=2MA−4MBR = 2M_A - 4M_BR=2MA−4MB.
Determine the probability that the residual mass is less than 15 grams, P(R<15)P(R < 15)P(R<15).
A third reagent, a catalyst CCC, has a mass distributed as C∼N(50,σ2)C \sim \text{N}(50, \sigma^2)C∼N(50,σ2). Three independent samples of the catalyst, C1,C2, and C3C_1, C_2, \text{ and } C_3C1,C2, and C3, are selected and combined with one sample each of reagents MAM_AMA and MBM_BMB.
The random variable TTT is defined as the total mass of the combined mixture: T=MA+MB+∑i=13CiT = M_A + M_B + \sum_{i=1}^3 C_iT=MA+MB+∑i=13Ci.
Given that the probability of the total mass exceeding 290 grams is 0.0228, and assuming all reagent masses are independent,
calculate the value of σ\sigmaσ, the standard deviation of the catalyst mass.
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.