In an ecological study of a specific butterfly population, it is found that 35% of the individuals carry a specific genetic marker, denoted by WWW. A researcher captures a random sample of 30 butterflies. Let the random variable XXX represent the number of butterflies in the sample who carry marker WWW.
Find (i) P(X=12)P(X = 12)P(X=12) (ii) P(X>8)P(X > 8)P(X>8) (iii) the smallest value of kkk such that P(X≤k)≥0.75P(X \le k) \ge 0.75P(X≤k)≥0.75
The study expands to a much larger region, and a new random sample of 400 butterflies is collected.
(i) Using a suitable normal approximation, find the probability that no more than 130 of these 400 butterflies carry marker WWW. (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.