An oceanographic research station utilizes two distinct models of deep-sea power probes, Alpha and Beta, for underwater data collection. The battery life, XXX hours, of a Model Alpha probe is normally distributed such that X∼N(120,25)X \sim N(120, 25)X∼N(120,25). The battery life, YYY hours, of a Model Beta probe is normally distributed such that Y∼N(128,16)Y \sim N(128, 16)Y∼N(128,16). The battery lives of the probes are independent.
Determine the probability that a Model Alpha probe, selected at random, outlasts a randomly selected Model Beta probe.
An engineer deploys 3 Model Alpha probes and 1 Model Beta probe.
Calculate the probability that the sum of the battery lives of the 3 Model Alpha probes is greater than 2.5 times the battery life of the Model Beta probe.
A composite metric SSS is defined as S=k1X+k2YS = k_1 X + k_2 YS=k1X+k2Y, where k1k_1k1 and k2k_2k2 are real constants. The engineer requires the mean of SSS to be exactly 3000 while ensuring the variance of SSS is minimized.
Determine the specific values of k1k_1k1 and k2k_2k2 that satisfy these requirements.
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.