A viticulturist is investigating the survival of a new hybrid vine graft. The probability that a graft is successful is denoted by ppp. A sample of 100 grafts is taken, and the random variable S S\,S represents the number of successful grafts, where S∼B(100,p)S \sim \text{B}(100, p)S∼B(100,p).
Using a normal approximation, the probability that S S\,S is at least 53 is 0.0054 to 4 decimal places.
Show that p p\,p satisfies 426.01p2−446.01p+110.25=0426.01p^2 - 446.01p + 110.25 = 0426.01p2−446.01p+110.25=0 when normal probability tables are used.
Hence find the value of ppp, justifying your answer.
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.