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Mathematics 9709/53 October/November 2020: question paper and mark scheme

Cambridge AS & A Level Mathematics (9709), Paper 5 Probability & Statistics 1, October/November 2020. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.

9709_w20_qp_539709_w20_ms_539709/53/O/N/20

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Paper
Paper 5 Probability & Statistics 1
Total marks
50
Time
1 hour 15 minutes
Code on the paper
9709/53/O/N/20
9709_w20_ms_53.pdf

9709/53 mark scheme, question by question

Each question as it's printed, and the mark scheme's points for it, as text. Diagrams, tables and graphs are in the PDFs above.

Question 16 marks

Question

The times taken to swim 100 metres by members of a large swimming club have a normal distribution with mean 62 seconds and standard deviation 5 seconds. (a) Find the probability that a randomly chosen member of the club takes between 56 and 66 seconds to swim 100 metres. [3] (b) 13% of the members of the club take more than t minutes to swim 100 metres. Find the value of t. [3]

Mark scheme

1(a) 56 − 62 66 − 62 M1 Using ± standardisation formula at least once, no σ or σ 2 , P(56 < X < 66) = P <z< 5 5 allow continuity correction = P( −1.2 < z < 0.8) Φ ( 0.8) + Φ (1.2) − 1 M1 Appropriate area Φ , from standardisation formula in final solution = 0.7881 + 0.8849 – 1 0.673 A1 3 1(b) z = 1.127 B1 ±(1.126 – 1.127) seen, 4 sf or more 60t − 62 = 1.127 M1 z-value = ± ( 60t − 62 ) condone z-value = ( t − 62 ) ± 5 5 5 60t = 5.635+62=67.635 no continuity correction, condone σ or σ 2 t = 1.13 A1 CAO 3

Question 25 marks

Question

An ordinary fair die is thrown until a 6 is obtained. (a) Find the probability that obtaining a 6 takes more than 8 throws. [2] Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable X . (b) Find the expected value of X . [1] (c) Find the probability that obtaining a pair of 6s takes either 10 or 11 throws. [2]

Mark scheme

2(a) 5 8 M1 p8, 0 < p < 1, no x, + or - 6 0.233 A1 2 2(b) 36 B1 1 2(c) 9 35 1 35 1 10 M1 OE, unsimplified expression in form p9 q + p10 q , P(X =10) + P(X=11) = + 36 36 36 36 p + q = 1, no × 0.0425 A1 2

Question 36 marks

Question

A committee of 6 people is to be chosen from 9 women and 5 men. (a) Find the number of ways in which the 6 people can be chosen if there must be more women than men on the committee. [3] The 9 women and 5 men include a sister and brother. (b) Find the number of ways in which the committee can be chosen if the sister and brother cannot both be on the committee. [3]

Mark scheme

3(a) Scenarios: M1 Correct number of ways for either 5 or 4 women, accept 6W 0M 9C6 = 84 unsimplified 5W 1M 9C5 × 5C1 = 126 × 5 = 630 4W 2M 9C4 × 5C2 = 126 × 10 = 1260 M1 Summing the number of ways for 2 or 3 correct scenarios (can be unsimplified), no incorrect scenarios. Total = 1974 A1 3 3(b) Total number of ways = 14C6 (3003) M1 14 C6 – a value Number with sister and brother = 12C4 (495) Number required = 14C6 – 12 12 C4 = 3003 – 495 M1 Cx or nC4 seen on its own or subtracted from their total, x ⩽ 6, n ⩽ 13 2508 A1 Alternative method for question 3(b) Number of ways with neither = 12C6 = 924 M1 12 C6 + a value 12 Number of ways with either brother or sister (not both) M1 Cx × 2 or nC5 × 2 seen on its own or added to their number of = 12C5 × 2 (= 792 × 2) = 1584 ways with neither, x ⩽ 5, n ⩽ 12 Number required = 924 + 1584 A1 = 2508 3

Question 48 marks

Question

The 13 00 train from Jahor to Keman runs every day. The probability that the train arrives late in Keman is 0.35. (a) For a random sample of 7 days, find the probability that the train arrives late on fewer than 3 days. [3] A random sample of 142 days is taken. (b) Use an approximation to find the probability that the train arrives late on more than 40 days. [5]

Mark scheme

4(a) 0.657 + 7C1 0.656 0.351 + 7C2 0.655 0.352 M1 Binomial term of form 7Cx p x (1 − p )7− x , 0 < p < 1, any p, x ≠ 0, 7 0.049022 + 0.184776 + 0.29848 A1 Correct unsimplified answer 0.532 A1 3 4(b) Mean = 142 ×0.35 = 49.7 B1 Correct unsimplified np and npq (condone σ = 5.684 evaluated) Variance = 142 × 0.35 × 0.65 = 32.305 40.5 − 49.7 M1 Substituting their µ and σ (no σ or σ 2 ) into ±standardisation P(X > 40) = P( z > ) 32.305 formula with a numerical value for '40.5' P( z > −1.619) M1 Using either 40.5 or 39.5 within a ±standardisation formula M1 Appropriate area Φ , from standardisation formula P(z >…) in final solution, must be probability 0.947 A1 Correct final answer 5

Question 57 marks

Question

The 8 letters in the word RESERVED are arranged in a random order. (a) Find the probability that the arrangement has V as the first letter and E as the last letter. [3] (b) Find the probability that the arrangement has both Rs together given that all three Es are together. [4]

Mark scheme

5(a) 8! B1 Correct unsimplified expression for total number of ways Total number of ways = (= 3360) 3!2! 6! B1 6! Number of ways with V and E in correct positions = (= 180) alone or as numerator in an attempt to find the number of 2!× 2! 2!× 2! ways with V and E in correct positions. No ×, ± 180 3 B1 FT 6! Probability = = or 0.0536 Final answer from their divided by their total number of 3360 56 2!× 2! ways Alternative method for question 5(a) 1 3 M1 a b × × seen, no other terms (correct denominators) 8 7 8 7 M1 1 3 × seen, no other terms (correct numerators) c d 3 A1 or 0.0536 56 3 5(b) Rs together and Es together: 5! (120) B1 Alone or as numerator of probability to represent the number of ways with Rs and Es together, no ×, +, – 6! B1 Alone or as denominator of probability to represent the number Es together: ( = 360 ) of ways with Es together, no ×, + or – 2! 5! M1 their 5! Probability = seen 6! 6! their 2! 2! 1 A1 OE 3 Alternative method for question 5(b) 5! 1 B1 P(Rs together and Es together): = their total number of ways 28 6! 3 B1 Alone or as numerator of probability to represent the P(Es together): 2! = P(Rs and Es together), no ×, +, – 28 their total number of ways 1 M1 Alone or as denominator of probability to represent the P(Es together), no ×, + or – Probability = 28 3 28 1 A1 1 their 3 OE, 28 seen 3 their 28 4

Question 68 marks

Question

Three coins A, B and C are each thrown once. ³ Coins A and B are each biased so that the probability of obtaining a head is 23 . ³ Coin C is biased so that the probability of obtaining a head is 54 . (a) Show that the probability of obtaining exactly 2 heads and 1 tail is 94 . [3] The random variable X is the number of heads obtained when the three coins are thrown. (b) Draw up the probability distribution table for X . [3] (c) Given that E X = 32 15 , find Var X . [2]

Mark scheme

6(a) Scenarios: M1 One 3 factor probability with 3, 3, 5 as denominators 2 2 1 4 HHT: × × = M1 3 factor probabilities for 2 or 3 correct scenarios added, no 3 3 5 45 2 1 4 8 incorrect scenarios HTH: × × = 3 3 5 45 1 2 4 8 THH: × × = 3 3 5 45 20 4 A1 AG, Total of 3 products with clear context Total = = 45 9 3 6(b) B1 Probability distribution table with correct outcomes with at least x 0 1 2 3 one probability, allow extra outcome values if probability of Prob. zero stated’ 1 8 20 16 45 45 45 45 B1 2 of P(0), P(1) and P(3) correct B1 FT 3 or 4 probabilities sum to 1 with P(2) correct 3 6(c) 02 × 1 + 12 × 8 + 22 × 20 + 32 × 16 32 2 M1 Substitute their attempts at scores in correct variance formula, Var(X) = − must have ‘– mean2’ (FT if calculated) (condone probs not 45 15 summing to 1); must be at least 2 non-zero values 2 8 80 144 32 = + + − 45 45 45 15 136 A1 or 0.604 225 2

Question 710 marks

Question

A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table. Number of incorrect notes 1−5 6 − 10 11 − 20 21 − 40 41 − 70 Frequency 10 5 26 32 18 (a) Draw a histogram to represent this information. [5] (b) State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range. [2] (c) Calculate an estimate for the mean number of incorrect notes. [3] 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.

Mark scheme

7(a) Class widths: 5, 5, 10, 20, 30 M1 At least 3 class widths correct and used in a calculation Frequency density: 2, 1, 2.6, 1.6, 0.6 M1 At least 3 correct frequency densities unsimplified – FT their class widths A1 All correct heights on a histogram using a linear vertical scale from zero – no FT B1 Correct upper bar ends (5.5, 10.5, 20.5, 40.5, 70.5) and 4 correct lower bar ends of 5.5, 10.5, 20.5, 40.5. Condone 0 or 1. B1 Linear scales with at least 3 values indicated on each axis, vertical scale from 0, axes labelled ‘fd’ and ‘no. of (incorrect) notes’, or better. 5 7(b) LQ: 11 – 20 B1 Both UQ and LQ correct UQ: 21 – 40 Greatest IQR = 40 – 11 = 29 B1 FT Subtract lower end of their LQ interval from upper end of their UQ interval 2 7(c) Midpoints: 3 8 15.5 30.5 55.5 M1 At least 4 midpoints correct and used 3 × 10 + 8 × 5 + 15.5 × 26 + 30.5 × 32 + 55.5 × 18 M1 Correct formula with their midpoints (not upper boundary, Mean = lower boundary, class width, frequency density, frequency or 91 30 + 40 + 403 + 976 + 999 cumulative frequency) = 91 2448 = 91 82 A1 Accept 26 or 27 26.9, 26 91 3

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Where can I find the 9709/53 October/November 2020 mark scheme?

On this page, free: the mark scheme for AS & A Level Mathematics 9709/53, October/November 2020 (9709_w20_ms_53), and the question paper (9709_w20_qp_53), to read or download, with each question's marking points written out underneath.

How many marks is 9709/53 out of?

Mathematics 9709/53 for October/November 2020 is out of 50 marks, with 1 hour 15 minutes to answer it.

Can I get my 9709/53 answers marked?

Yes. Upload your answers to 9709/53/O/N/20 as a PDF or photos, and CIE Buddy marks every question against this paper's official mark scheme, using Cambridge's marking rules. You get your score, each lost mark explained on your own script, and the topics to revise. It's a guide, not an official Cambridge grade.