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4 A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by p. (a) Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of p, for the variance of the number of green beads Sasha selects. [1] Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the X th bead that Freda selects. (b) Assume that p = 0.3. Find (i) P(X > 5), [2] (ii) Var (X). ] (c) In fact, on the basis of a large number of observations of X, it is found that P(X = 3) = 75 XP(X= 1). Estimate the value of p. [5]1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution Po(120). (a) Find the probability that at least 28 orders are received in a randomly chosen 10-minute period. [2] (b) Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received. [3] (c) State a necessary assumption for the validity of your calculation in part (b). [1] 2 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women. (a) Find the probability that all the men are next to each other. [3] (b) Find the probability that no two men are next to one another. [4] 3 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics. (a) For all sixteen candidates, the value of the product moment correlation coefficient r for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the 5% significance level, of association between the marks on the two papers. [4] (b) A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by n = 10, Ex = 750, Ey = 690, Ex? = 57 690, Zy = 49676, Exy = 50829. (i) Calculate the value of r for these 10 candidates. [2] (ii) What do the two values of r, in parts (a) and (b)(i), tell you about the scores of the sixteen candidates? [1]