In a large city, 45% of residents use public transport for their daily commute. A random sample of 40 residents is taken. Let the random variable XXX represent the number of residents in the sample who use public transport.
Find (i) P(X=22)P(X = 22)P(X=22) (ii) P(X<15)P(X < 15)P(X<15) (iii) the smallest value of kkk such that P(X≤k)>0.6P(X \le k) > 0.6P(X≤k)>0.6
A new larger random sample of 500 residents is taken.
(i) Using a suitable normal approximation, find the probability that at least 215 of these 500 residents use public transport. (ii) Justify why a normal approximation is appropriate in part (b)(i).
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.