A geomechanical boring tool is used to create pilot holes for foundation piles. The depth, DDD millimetres, achieved in a single standard operation of the tool is normally distributed with mean 500500500 and standard deviation 202020.
Using standardisation, show that the probability that an operation results in a depth of less than 492 mm492 \text{ mm}492 mm is 0.34460.34460.3446.
A specific site requirement necessitates a hole depth of at least 492 mm492 \text{ mm}492 mm. A technician is permitted up to 3 independent attempts at a single location to meet this requirement. If an attempt meets or exceeds the requirement, no further attempts are made at that location.
Given that the technician eventually successfully met the requirement at a location, determine the probability that success was achieved on the second attempt with a hole deeper than 530 mm530 \text{ mm}530 mm.
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.