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A Level Maths: Probability & Statistics 1 Exam Paper
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Cambridge International AS & A Level CANDIDATE NAME CENTRE NUMBER CANDIDATE NUMBER *0009306270* MATHEMATICS 9709/52 Paper 5 Probability & Statistics 1 May/June 2021 1 hour 15 minutes You must answer on the question paper. You will need: List of formulae (MF19) INSTRUCTIONS ! Answer all questions. ! Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ! Write your name, centre number and candidate number in the boxes at the top of the page. ! Write your answer to each question in the space provided. ! Do not use an erasable pen or correction fluid. ! Do not write on any bar codes. ! If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown. ! You should use a calculator where appropriate. ! You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. ! Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION ! The total mark for this paper is 50. ! The number of marks for each question or part question is shown in brackets [ ]. This document has 12 pages. JC21 06_9709_52/2R © UCLES 2021 [Turn over 2 1 ppl f 5 An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable X . (a) Write down the mean of X . [1] Xv Gt ........................................................................................................................................................ E x I 6 ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw. [2] P PH P 4 P5 P 6 ........................................................................................................................................................ 18148 E to 18547 8 4 ........................................................................................................................................................ ........................................................................................................................................................ 0 2996 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Find the probability that a 5 is first obtained in fewer than 10 throws. [2] ........................................................................................................................................................ Pl x Cio PIX 29 1 I 1 8 ........................................................................................................................................................ ........................................................................................................................................................ 0.806 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 3 2 The weights of bags of sugar are normally distributed with mean 1.04 kg and standard deviation " kg. In a random sample of 2000 bags of sugar, 72 weighed more than 1.10 kg. Find the value of ". [4] X 04,82 ................................................................................................................................................................ 1 N ................................................................................................................................................................ P X 1.10 72 0 0.036 ................................................................................................................................................................ ................................................................................................................................................................ this ................................................................................................................................................................ É Z................................................................................................................................................................ 0.9647 1.798 ................................................................................................................................................................ 04 1.798 84 798 ................................................................................................................................................................ i 11031 ................................................................................................................................................................ 8 0.0334 ................................................................................................................................................................ i ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 [Turn over z 4 3 On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are 0.4, 0.35 and 0.25 respectively. When she travels by bus, the probability that she arrives late is 0.55. When she travels by train, the probability that she arrives late is 0.7. When she travels by car, the probability that she arrives late is x. On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late is 0.48. (a) Find the value of x. [3] 0.55 L ........................................................................................................................................................ I B ........................................................................................................................................................ o 45 0.4 ........................................................................................................................................................ L 0.7 0.35 ........................................................................................................................................................ y L 0.25 ........................................................................................................................................................ x L 0.3 ........................................................................................................................................................ e i ........................................................................................................................................................ ........................................................................................................................................................ 0.4 0.45 0.35 0.3 0.2511 X 0 48 ........................................................................................................................................................ 0.2511 0.195 x ........................................................................................................................................................ 1 0.22 (b) Find the probability that Alexa travels to work by train given that she arrives late. [3] ........................................................................................................................................................ P 0.35 0.7 44 0.471 ........................................................................................................................................................ l 0.48 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 5 4 A fair spinner has sides numbered 1, 2, 2. Another fair spinner has sides numbered −2, 0, 1. Each spinner is spun. The number on the side on which a spinner comes to rest is noted. The random variable X is the sum of the numbers for the two spinners. (a) Draw up the probability distribution table for X . [3] ........................................................................................................................................................ 2 2 I O L O 1 2 3 X ........................................................................................................................................................ O 2 1 ........................................................................................................................................................ 4 4 43 Pax ........................................................................................................................................................ 1 o 22 ........................................................................................................................................................ I 2 3 3 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find E!X " and Var!X ". [3] ........................................................................................................................................................ ........................................................................................................................................................ 0 Fix 43 69 4 ........................................................................................................................................................ 4 4 ........................................................................................................................................................ ........................................................................................................................................................ varas 4 04 4 9 155 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Varix ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 [Turn over 6 5 Every day Richard takes a flight between Astan and Bejin. On any day, the probability that the flight arrives early is 0.15, the probability that it arrives on time is 0.55 and the probability that it arrives late is 0.3. (a) Find the probability that on each of 3 randomly chosen days, Richard’s flight does not arrive late. [1] P Not late 1 0.3 0.7 ........................................................................................................................................................ ........................................................................................................................................................ 6.7 Probability 0.343 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the probability that for 9 randomly chosen days, Richard’s flight arrives early at least 3 times. [3] 0.157 Xu B 9 ........................................................................................................................................................ ........................................................................................................................................................ PIX 73 1 Plo Pal Plz ........................................................................................................................................................ ........................................................................................................................................................ I 10.859 910.15 0.853 9510.10851 ........................................................................................................................................................ ........................................................................................................................................................ 0.141 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 7 (c) 60 days are chosen at random. Use an approximation to find the probability that Richard’s flight arrives early at least 12 times. [5] Xv B 60 0.15 60,8 0.15 1 0.85 n ........................................................................................................................................................ ........................................................................................................................................................ M Np 60 0.15 9 8Enpq 7.65 ........................................................................................................................................................ ........................................................................................................................................................ Xv N 9,7 65 75 ........................................................................................................................................................ 8 ........................................................................................................................................................ I ........................................................................................................................................................ ........................................................................................................................................................ I ........................................................................................................................................................ 8 404 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ P 1 810.904 12 x 11.51 X........................................................................................................................................................ P ........................................................................................................................................................ I 0.8169 ........................................................................................................................................................ 0.183 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 [Turn over 8 6 (a) Find the total number of different arrangements of the 8 letters in the word TOMORROW. M T [2] W ........................................................................................................................................................ ........................................................................................................................................................ 8 Rr O ........................................................................................................................................................ ........................................................................................................................................................ number 3360 of arrangements 38,4 21 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the total number of different arrangements of the 8 letters in the word TOMORROW that have an R at the beginning and an R at the end, and in which the three Os are not all together. [3] T 8 Dad M RR W ........................................................................................................................................................ ........................................................................................................................................................ 0 ofarrangements with R at beginingand Rat END ........................................................................................................................................................ and no restrictions for theletters ni between ........................................................................................................................................................ Number ........................................................................................................................................................ R T OO O M W R ........................................................................................................................................................ 120 arrogent ........................................................................................................................................................ Sf ........................................................................................................................................................ agents with R at begining and Rat END ........................................................................................................................................................ and the three O's me together Nuba of ........................................................................................................................................................ arr ........................................................................................................................................................ R OOO T M W R ........................................................................................................................................................ ........................................................................................................................................................ 24 arrangent 4 ........................................................................................................................................................ Wanted number ofarrangement 120 24 96 ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 9 Four letters are selected at random from the 8 letters of the word TOMORROW. spa (c) Find the probability that the selection contains at least one O and at least one R. Hard [5] ........................................................................................................................................................ 187 O RR ........................................................................................................................................................ ........................................................................................................................................................ O R others ........................................................................................................................................................ 18 2 I 39 29 35 1 2 I ........................................................................................................................................................ I 9 39 28 30 1 2 ........................................................................................................................................................ I 29 34 18 ........................................................................................................................................................ 39 O ........................................................................................................................................................ 2 2 3 36 29 38 1 O ........................................................................................................................................................ 2 3 x 36 3g 29 ........................................................................................................................................................ selection 50 ........................................................................................................................................................ ........................................................................................................................................................ Probability 581 57 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 [Turn over 10 7 The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are shown below. Amazons 205 198 181 182 190 215 201 178 202 196 184 Giants 175 182 184 187 189 192 193 195 195 195 204 (a) State an advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to illustrate this information. [1] It includes all the data ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the left-hand side of the diagram. [4] Giants Amazons 4 8 5 8 17 2 1 18 2 6 0 19 2 2 1 20 4 5 4 3 7 5 5 21 Key 111812 means and © UCLES 2021 181 an for Amazons 1820h for Giants 9709/52/M/J/21 9 5 5 11 (c) Find the interquartile range of the heights of the players in the Amazons. [2] ........................................................................................................................................................ 29 182 40 202 ........................................................................................................................................................ ........................................................................................................................................................ IQR 202 182 20 An ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Four new players join the Amazons. The mean height of the 15 players in the Amazons is now 191.2 cm. The heights of three of the new players are 180 cm, 185 cm and 190 cm. (d) Find the height of the fourth new player. [3] ........................................................................................................................................................ 182 1841190 196 1984201 2021205 215 180485 190tx 178 181........................................................................................................................................................ ........................................................................................................................................................ 15 ........................................................................................................................................................ 3 191.2 ........................................................................................................................................................ ........................................................................................................................................................ I ........................................................................................................................................................ 2687 X 191.2 15 ........................................................................................................................................................ ........................................................................................................................................................ X 181 An ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2021 9709/52/M/J/21 12 Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. © UCLES 2021 9709/52/M/J/21
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