OCR 2023 GCE FURTHER MATHEMATICS B MEI Y412/01: STATISTICS A AS LEVEL QUESTION PAPER & MARK SCHEME (MERGED)

Ryan has 6 one-pound coins and 4 two-pound coins. Ryan decides to select 3 of these coins at random to donate to a charity. The total value, in pounds, of these 3 coins is denoted by the random variable X. (a) Show that P(X = 3) = 1 . [2] The table below shows the probability distribution of X. r 3 4 5 6 P(X = r) 1 6 1 2 3 10 1 30 (b) Draw a graph to illustrate the distribution. [2] (c) In this question you must show detailed reasoning. Find each of the following. • E(X) • Var(X) [5] Ryan’s friend Sasha decides to give the same amount as Ryan does to the charity plus an extra three pounds. The random variable Y represents the total amount of money, in pounds, given by Ryan and Sasha. (d) Determine each of the following. • E(Y) • Var(Y) [3] 3 2 A group of friends live by the sea. Each day they look out to sea in the hope of seeing a dolphin. The probability that they see a dolphin on any day is 0.15. You should assume that this probability is not affected by whether or not they see a dolphin on any other day. (a) Explain why you can use a geometric distribution to model the number of days that it takes for them to first see a dolphin. [1] (b) Find the probability that they see a dolphin for the first time on the fifth day. [1] (c) Find the probability that they do not see a dolphin for at least 10 days. [1] (d) Determine the mean and the variance of the number of days that it takes for them to see a dolphin. [3] 3 At a pottery which manufactures mugs, it is known that 5% of mugs are faulty. The mugs are produced in batches of 20. Faults are modelled as occurring randomly and independently. The number of faulty mugs in a batch is denoted by the