Cambridge A Level Mathematics 9709 — 2002 Oct/Nov Paper 6 · Variant 1

CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level General Certificate of Education Advanced Level Advanced International Certificate of Education MATHEMATICS STATISTICS 9709/6 0390/6 PAPER 6 Probability & Statistics 1 (S1) OCTOBER/NOVEMBER SESSION 2002 1 hour 15 minutes Additional materials: Answer paper Graph paper List of Formulae (MF9) TIME 1 hour 15 minutes INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 50. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This question paper consists of 3 printed pages and 1 blank page. CIE 2002 [Turn over

2 1 The discrete random variable X has the following probability distribution. x 1 3 5 7 P(X
x) 0.3 a b 0.25 (i) Write down an equation satisfied by a and b. [1] (ii) Given that E(X)
4, find a and b. [3] 2 Ivan throws three fair dice. (i) List all the possible scores on the three dice which give a total score of 5, and hence show that the probability of Ivan obtaining a total score of 5 is 1 36. [3] (ii) Find the probability of Ivan obtaining a total score of 7. [3] 3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m. (i) Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m. [3] (ii) The school gives a certificate to the 10% of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate. [3] 4 In a certain hotel, the lock on the door to each room can be opened by inserting a key card. The key card can be inserted only one way round. The card has a pattern of holes punched in it. The card has 4 columns, and each column can have either 1 hole, 2 holes, 3 holes or 4 holes punched in it. Each column has 8 different positions for the holes. The diagram illustrates one particular key card with 3 holes punched in the first column, 3 in the second, 1 in the third and 2 in the fourth. (i) Show that the number of different ways in which a column could have exactly 2 holes is 28. [1] (ii) Find how many different patterns of holes can be punched in a column. [4] (iii) How many different possible key cards are there? [2] 9709/6/O/N/02

3 5 Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6. If Rachel wins a particular game, the probability of her winning the next game is 0.7, but if she loses, the probability of her winning the next game is 0.4. By using a tree diagram, or otherwise, (i) find the conditional probability that Rachel wins the first game, given that she loses the second, [5] (ii) find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play. [4] 6 (i) A manufacturer of biscuits produces 3 times as many cream ones as chocolate ones. Biscuits are chosen randomly and packed into boxes of 10. Find the probability that a box contains equal numbers of cream biscuits and chocolate biscuits. [2] (ii) A random sample of 8 boxes is taken. Find the probability that exactly 1 of them contains equal numbers of cream biscuits and chocolate biscuits. [2] (iii) A large box of randomly chosen biscuits contains 120 biscuits. Using a suitable approximation, find the probability that it contains fewer than 35 chocolate biscuits. [5] 7 The weights in kilograms of two groups of 17-year-old males from country P and country Q are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, . . . 4
7
1 . . . denotes weights of 74 kg for a male in country P and 71 kg for a male in country Q. Country P Country Q 5 1 5 6 2 3 4 8 9 8 7 6 4 7 1 3 4 5 6 7 7 8 8 9 8 8 6 6 5 3 8 2 3 6 7 7 8 8 9 7 7 6 5 5 5 4 2 9 0 2 2 4 5 4 4 3 1 10 4 5 (i) Find the median and quartile weights for country Q. [3] (ii) You are given that the lower quartile, median and upper quartile for country P are 84, 94 and 98 kg respectively. On a single diagram on graph paper, draw two box-and-whisker plots of the data. [4] (iii) Make two comments on the weights of the two groups. [2] 9709/6/O/N/02

CAMBRIDGE INTERNATIONAL EXAMINATIONS NOVEMBER 2002 GCE Advanced Subsidiary Level Advanced International Certificate of Education UNIVERSITY of CAMBRIDGE Local Examinations Syndicate

Page 1 Mark Scheme Syllabus Paper AS Level & AICE Examinations — November 2002 9709, 0390 6 1 (@) a+b=0.45 Bl Accept unsimplified equation (ii) 0.34+3a4+5b4+7x0.25=4 MI For an equation involving YP: =4 must be MI correct unsimplified version, seen anywhere For sensible attempt to solve the two equations ie eliminati letter a=015 b=03 Al ee For correct'a and b. For an option involving(1,2,2) and an option involving (1,1,3) For all six correct options For legitimately obtaining answer given 2 (i) options (122), (212), (221), (113), | M1 (131), (311) prob = 6 /216 (AG) Gi) (133)x3, (223)x3, (115) x 3, For listing 3 or 4 different correct options or tree (124)x 6 diagram For multiplying 4 prob options by a relevant number or listing = 12 correct options prob = 15/216 (= 5/72) . For correct answer 3) z =440735-0 _ 49.431 For standardising (JiL6 in denom Mi, ccMO 16 11.6 M0) For subtracting two relevant probabilities or equivalent (0.431) - {1 - B(0.431)} = 0.334 For correct answer (ii) z = + 1.282 or +1.281 only For stating z 1.282=22 35.0 TG For solving an equation for x with some z value from tables, allow cc, 11.6 , 35-x, not 11.67 x= 49.9 or 49.8 on z= 1.28 For correct answer 4 (i) Cy = 28 oF 7464544434241 For sC; For listing 4 Combination options (can be added or multiplied here) For gC; + Cz + sC3+ Cy For at least 3 correct numbers, can be implied by seeing 878080 (mult) For correct answer SR gCrtgCz+...+3Cs MI only SR gC3x gC3xsC xsCz MI only (ii) ei + sy + Cs + 2Cy =8+28+56+70 = 162 (ii) (162) = 688 747 536 or 3s For (their (ii))* or ¢C3+gCstCi x sCo For correct answer in any form

Mark Scheme Syllabus Paper F Page 2 AS Level & AICE Examinations —- November 2002 9709, 0390 5@ PW, | Ly For 0.60.3 seen anywhere in isolation For correct numerator = MI For summing two 2 factor products in denom _ Al For correct denominator unsimplified _ Al___5 | For correct answer (ii) P(W1W2L3) = 0.6x0.7x 0.3 M1 For summing three probability options P(WiL2W3) = 0.6x 0.3x 0.4 Bl For one correct probability option P(L;W2W3) = 0.4x 0.4x 0.7 Bl For two correct probability options Probability = 0.31 Al 4 | For correct answer 6 (i) P(equal) = (0.25) x (0.75)*x Cs M1 For (0.25)°x (0.75)° must be 0.25, 0.75 0.0584 Al 2 | For correct answer. AO if subsequently doubled (ii) (0.0584)! (0.9416)"x sCr M1 For (their(a))'x (1 - their(a))’ xsC1 = 0.307 Alft 2 | For correct answer from their ans to (i) Accept anything from 0.304 to 0.307 for the ft if they have lost the Al in (i) from PA (iii) p = 120 0.25 = 30, 0” = 30 x 0.75= 22.5 | M1 For both mean and variance correct from any sensible p M1 For correct standardisation with or without cc POr<35)= af (0.949) Bl For correct use of continuity correction 34.5 MI For use of tables based on their z value either = 0.829 end NB can’t get if z is too large or too small Al__5 | For correct answer 7 @) LQ = 72, or 73 or 71.5 only Bl Accept Qi, Qo, Qs median = 78, Bl LQ UQ muddle scores BI BO and possibly B1 for UQ = 88 or 87.75 only BI 3, | median (i) Bl For only one numbered linear scale P —{_T}+- Bl For country P all correct on linear scale BIft For Q all correct on linear scale Q Bl 4 | For P and Q labelled, weights or kg shown SR non linear scale max BO BO BO B1 5p—_6p__7)__80_9h 10 110 ws Or max BO Bl BO BI if one error in an otherwise linear scale . ee _ NB No outliers (iii) people heavier in P than in Q “TBL Or equivalent statement weights more spread out in Q Bl 2. | Or equivalent statement Cannot have two statements saying the equivalent of the same category (wts, spread, skewness). Must have the same statement relating to P and to Q.