A service elevator has a maximum load capacity of 900 kg900\text{ kg}900 kg.
The weights of adult males are normally distributed with mean 82 kg82\text{ kg}82 kg and standard deviation 12 kg12\text{ kg}12 kg.
The weights of adult females are normally distributed with mean 68 kg68\text{ kg}68 kg and standard deviation 8 kg8\text{ kg}8 kg.
Assume that the weights of individual people are independent.
Find the probability that when 7 males7\text{ males}7 males and 4 females4\text{ females}4 females are in the elevator, the total load exceeds 900 kg900\text{ kg}900 kg.
The elevator displays a sign stating: "Maximum occupancy is ccc persons."
Find the largest integer value of ccc such that the probability of the total load exceeding 900 kg900\text{ kg}900 kg is less than 1%1\%1%, regardless of the gender distribution of the occupants.
390 exam-style questions on OCR A Level Maths 2.4 Statistical Distributions, covering 2.4.1 Discrete probability distributions, 2.4.2 Binomial distribution as a model, 2.4.3 Calculating binomial probabilities, 2.4.4 Mean and variance of the binomial (A-level only), 2.4.5 Normal distribution as a model (A-level only), 2.4.6 Probabilities using the normal distribution (A-level only), 2.4.7 Links to histograms, mean and standard deviation (A-level only), and 2.4.8 Selecting an appropriate distribution (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.