S1 Arrangements And Combinations.pdf

OCR Maths S1 Topic Questions from Papers Arrangements and Combinations


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(iii) On Wednesday 8 boxes are selected, and on Thursday another 8 boxes are selected. Find the probability that on one of these days the number of boxes containing fewer than 42 matches is 0, PhysicsAndMathsTutor.com and that on the other day the number is 2 or more. [3]

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An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions. (i) If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions. [3]

Four of thedown questions, including geometric distributions, are worth 7 marks each, and [4] the (i) Write the values of a, the b, cone , d, eonand f. remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7-mark the discs first four questions thedenoted paper, but arranged random order. The . The table in shows the probability The totalquestions number are of red chosen, out of on 3, is by Rare 9-mark questions are the last four questions, but are arranged in random order. Find the probability distribution of R. that r 0 1 2 3 (ii) the questions on geometric distributions and on binomial distributions are next to one another, physicsandmathstutor.com 9 1 1 [3] P(R = r) k 10

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2 on binomial distributions are separated by at least 2 (iii) the questions on geometric distributions and other questions. [4] 9 Some observations of bivariate data (ii) Show how to obtain the value P(were R = 2)made = 20 .and the equations of the two regression lines were [3] (Q8, Jan 2005) foundobservations to be as follows. Five of bivariate data produce the following results, denoted as (xi , yi ) for i = 1, 2, 3, 4, 5. (iii) Find the value of k. [2] y on x : y = −0.6x + 13.0 (13, 2.7) (13, x4.0) (18, 2.8) (23, 3.3) (23, 2.2) on y : x = −1.6y + 21.0 (iv) Calculate the mean and variance of R2. [5] 2 x = 90, Σ y = 15.0, Σ x = 1720, Σ y = 46.86, Σ xy = 264.0.] [Σ (i) State, with a reason, whether the correlation between x and y is negative or positive. [1]

A(ii) committee of the 7 people is to be chosen from 18 volunteers. (i) Show regression line of y at onrandom x an hasestimate gradient andoffind its equation Neitherthat variable is controlled. Calculate of −0.06, the value x when y = 7.0. in the form [2] y = a + bx. [4] (i) In how many different ways can the committee be chosen? [2] (iii) Findregression the valuesline of xisand y. to estimate the value of y corresponding to x = 20, but the value x =[3] (ii) The used 20

is accurate only to theofnearest whole Calculate difference the largestThe and The 18 volunteers consist 5 people fromnumber. Gloucester, 6 fromthe Hereford and between 7 from Worcester. take. the smallest that the estimated value of y couldthat committee is to 5bevalues chosen randomly. the probability the willthe bag. If it is red[3] A bag contains black discs and 3 redFind discs. A disc is selected at committee random from it is(ii) replaced it is not , ebag. , e ,Ifeit , is e black, are defined byreplaced. A second disc is now selected at random from The numbers consistinofethe 12 people 2 3 from 4 5 Gloucester, 2 people from Hereford and 3 people from Worcester, [4] the bag. = a Worcester, + bxi − yi for i = 1, 2, 3, 4, 5. (iii) include exactly 5 peopleefrom [4] i Find the probability that (iv) include at least 2 people from each of the three cities. [4] (iii) The values of e1 , e2 and e3 are 0.6, −0.7 and 0.2 respectively. Calculate the values of e4 and e5 . (i) the second disc is black, given that the first disc was black, [1] (Q7, June 2005) [2] 4732/S05 (ii) Calculate the secondthe discvalue is black, (iv) of e2 + e2 + e2 + e2 + e2 and explain the relevance of this quantity to [3] the 1

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lineare found in part (i) . (iii) regression the two discs of different colours.

[2] [3]

(v) Find the mean and the variance of e1 , e2 , e3 , e4 , e5 .

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[4]

Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a 4732/Jan05 row. (i) How many different arrangements of the letters are possible?

[3]

(ii) In how many of these arrangements are all three Ds together?

[2]

The 7 cards are now shuffled and 2 cards are selected at random, without replacement. (iii) Find the probability that at least one of these 2 cards has D printed on it.

[3]

(Q3, June 2006) 4

(i) The random variable X has the distribution B(25, 0.2). Using the tables of cumulative binomial [2] probabilities, or otherwise, find P(X ≥ 5).

(ii) The random variable Y has the distribution B(10, 0.27). Find P(Y = 3).

[2]

(iii) The random variable Z has the distribution B(n, 0.27). Find the smallest value of n such that [3] P(Z ≥ 1) > 0.95. 5

The probability distribution of a discrete random variable, X , is given in the table.

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The table shows the probability distribution for a random variable X . (ii) Give a reason why it would not be sensible to use your answer to draw a conclusion about all the PhysicsAndMathsTutor.com households in the town. x [1] 0 1 2 3 PX  x

0.1

0.2

0.3

0.4

The digits 1, 2, 3, 4 and 5 are arranged in random order, to form a five-digit number. Calculate EX  and VarX . (i) How many different five-digit numbers can be formed?

[5] [1]

Two placed skaters five countries in rank order. (ii) judges Find theeach probability that thefrom five-digit number is physicsandmathstutor.com (a) odd,

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[2] Position 1st 2nd4 3rd 4th 5th (b) less than 23 Judge 000. 1 [3] UK France Russia Poland Canada The stem-and-leaf diagram shows the masses, in grams, of 23 plums, measured correct to the nearest (Q3, Jan 2007) Judge 2 Russia Canada France UK Poland gram.

6 7 8 8 coefficient, 9 Calculate Spearman’s 5 rank5correlation rs , for the two judges’ rankings. [5] 6 1 2 3 5 6 8 9 Key : 6 2 means 62 7 0 0 2 4 5 6 7 8 0 8 (i) How many different teams of 7 people can be chosen, without regard to order, from a squad 9 7 of 15? [2] (ii) consists 6 forwards and of9 these defenders. (i) The Find squad the median and of interquartile range masses.How many different teams containing [3] 3 forwards and 4 defenders can be chosen? [2]

asJune a measure (ii) State one advantage of using the interquartile range rather than the standard deviation (Q3, 2007) [1] of the variation in these masses. 4 A bag contains 6 white discs and 4 blue discs. Discs are removed at random, one at a time, without physicsandmathstutor.com replacement. (iii) State one advantage and one disadvantage of using a stem-and-leaf diagram rather than a box2 [2] and-whisker plot to represent data. (i) Find the probability that (i) The letters A, B, calculate C, D and the E are arranged in a straight line. of the given data. He first subtracted 61 James wished mean andthe standard deviation (iv) (a) the secondtodisc is blue, given that first disc was blue, [1] 5 from each of the digits toarrangements the left of theare line in the stem-and-leaf diagram, giving the following. (a) How many different possible? [2] (b) the second disc is blue, [3] (b) In how many0of 5these are the letters A and B next to each other? [3] 6 7arrangements 8 8 9 (c) the third disc is blue, given that the first disc was blue. [3] 1 1 2 3 5 6 8 9 (ii) From the letters A,2 B, 0C,0D 2and letters areKey selected Find the probability 4 E, 5 two 6 7different 8 : 1 at2random. means 12 (ii) The random variable X is the number 4732/01 of discs © OCR 2007 Jan07 which are removed up to and including the first that these two letters are A and B. [2] 3 0 blue disc. State whether the variable X has a geometric distribution. Explain your answer briefly. 4 7 [1] (Q1, Jan 2008) 2

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1 in this diagram are 18.1 and 9.7 respectively, correct The mean and T standard of the data A random variable has thedeviation distribution Geo! ". Find 5 to 1 decimal place. Write down the mean and standard deviation of the data in the original (i) P( T = 4), [2] diagram.

(ii) P(T > 4),

A testE(consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J. (iii) T ).

[2]

[1]

The examiner plans to arrange all 8 questions in a random order, regardless of topic. A sample of bivariate data was taken and the results were summarised as follows. (i) (a) How many different arrangements are possible? [2] n=5 Σ x = 24 Σ x2 = 130 Σ y = 39 Σ y2 = 361 Σ xy = 212 (b) Find the probability that no two Algebra questions are next to each other and no two [3] Geometry questions are next to each other. (i) Show that the value of the product moment correlation coefficient r is 0.855, correct to 3 significant [2] Later,figures. the examiner decides that the questions should be arranged in two sections, Algebra followed

by Geometry, with the questions in each section arranged in a random order. (ii) The ranks of the data were found. One student calculated Spearman’s rank correlation coefficient found that rs = 0.7. Another student calculated the product moment coefficient, R,[2] of s , and (ii) r(a) How many different arrangements are possible? © OCR 2007 4732/01 Jun07 these ranks. State which one of the following statements is true, and explain your answer briefly. (b) Find the probability that questions A and H are next to each other. [1] (A) R = 0.855 (c)(B)Find R =the 0.7probability that questions B and J are separated by more than four other questions. [4] (C) It is impossible to give the value of R without carrying out a calculation using the original (Q6, Jan 2009) data. [2] (iii) All the values of x are now multiplied by a scaling factor of 2. State the new values of r and r .

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Three letters are selected at random from the 8 letters of the word COMPUTER, without regard to order. (i) Find the number of possible selections of 3 letters.

[2]

(ii) Find the probability that the letter P is included in the selection.

[3]

Three letters are now selected at random, one at a time, from the 8 letters of the word COMPUTER, and are placed in order in a line. (iii) Find the probability that the 3 letters form the word TOP.

[3]

(Q7, June 2009) 8

A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes countersphysicsandmathstutor.com at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white 5 counters have been taken. Niko plays the game once.

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The the word NEVER arranged in random order in a straight line. (i) five (a) letters Copy of and complete the treeare diagram showing the probabilities for Niko.

[4]

(i) How many different orders of the letters are possible?

[2]

(ii) In how many of the possible orders are the two Es next to each other?

[2]

First counter

19 the firstWhite two letters in the order include exactly one letter E. (iii) Find the probability that 20

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(Q8, Jan 2010)

R and S are independent random variables each having the distribution Geo(p). physicsandmathstutor.com 1 (i) Find P(R = 1 and S =201) in termsRed of p. 4

[1]

(ii) Show that P(R = 3 and S = 3) = p2 q4 , where q = 1 − p. The menu below all thethat dishes available certain restaurant. (b) Find theshows probability Niko will winataaprize.

[1] [3]

(iii) Use the formula for the sum to infinity of a geometric series to show that Rice dishes Main dishes by X . Vegetable dishes that Niko takes is denoted (ii) The number of counters p P(R Chicken = S) = . (a) Find P(X = 3). Boiled rice Mushrooms 2−p Beef Cauliflower (b) Find E(X ). Fried rice

[5] [2]

Pilau rice

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Lamb

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Spinach

Repeated independentKeema trials ofrice a certain experiment On each trial the probability of Mixed grillare carried out. Lentils success is 0.12. Prawn Potatoes (i) Find the smallest value of n such thatVegetarian the probability of at least one success in n trials is more [3] than 0.95.

A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes [5] (ii) Find the probability that the 3rd success occurs on the 7th trial. and 4 different vegetable dishes from this menu. Given these restrictions, (i) find the number of possible combinations of dishes that they can choose to share,

[3]

(ii) assuming that all choices are equally likely, find the probability that they choose boiled rice. Copyright Information

[2]

OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.

The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.

If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.

(iii) Find the number of possible combinations of dishes that they can now choose.

For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1PB.

[3]

OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

(Q7, June 2010)

© OCR 2009 4732 Jun09 The proportion of people who watch West Street on television is 30%. A market researcher interviews 8 people at random in order to contact viewers of West Street. Each day she has to contact a certain number of viewers of West Street.

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Red (i) The diagram shows67 cards, each with a digit printed on it. The digits form a 7-digit number. 10

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How many different 7-digit numbers can be formed using these cards?

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(ii) The diagram below4shows 5 white cards and 10 grey cards, each with a letter printed on it. 10

A

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Blue

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The total number of blue discs that Chloe takes is denoted by X .

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(ii) Show that P(X = 1) = 35 .

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From these cards, 3 white cards and 4 grey cards are selected at random without regard to order. The complete probability distribution of X is given below. (a) How many selections of seven cards are possible? [3] x 0 1 2 (b) Find the probability that the seven cards include exactly one card showing the letter A. [4] 7 1 3 P(X = x) 30 6 5 (Q6, Jan 2011) The probability distribution of a discrete random variable, X , is shown below. (iii) Calculate E(X ) and Var(X ).

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P

x

P(X = x)

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a A group of 7 students sit in random order on a bench.

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[5]

1−a

(i) Find E(X ) in terms of a. (i) (a) Find the number of orders in which they can sit.

[2] [1]

that7Var (X ) = 4include a(1 − aTom ). and Jerry. Find the probability that Tom and Jerry sit next[3] (ii) Show students to (b) The each other. [3] 8

Five dogs, A, B, C , D and E, took part in three races. The order in which they finished the first race The students consist of 3 girls and 4 boys. Find the probability that (ii) ABCDE was . (a) no two boys sit next to each other, [2] (i) Spearman’s rank correlation coefficient between the orders for the 5 dogs in the first two races was to be −1.sitWrite down order in which the dogs finished the second race. [1] all three girls next to eachthe other. [3] (b) found

(Q6,race June and2011) the (ii) Spearman’s rank correlation coefficient between the orders for the 5 dogs in the first third race was found to be 0.9. (a) Show that, in the usual notation (as in the List of Formulae), Σd 2 = 2.

© OCR 2011

[2]

(b) Hence or otherwise find a possible order in which the dogs could have finished the third race. [2] 4732 Jun11

Copyright Information OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © OCR 2011

4732 Jan11

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(b) Hence findProduct the probability the number of and these produce blue is (IMR), greater The Gross Domestic per Capitathat (GDP), x dollars, theplants Infantthat Mortality Rate perflowers thousand PhysicsAndMathsTutor.com than the number that produce red flowers. [3] y, of 6 African countries were recorded and summarised as follows. / x = 7000 / x 2 = 8 700 000 / y = 456 n= 6 A bag contains 9 discs numbered 1, 2, 3, 4, 5, 6, 7, 8, 9.

/ y 2 = 36 262

/ xy = 509 900

(i) Calculate the equation of the regression line of y on x for these 6 countries. (i) Andrea chooses 4 discs at random, without replacement, and places them in a row.

[4]

The original data were plotted on a scatter diagram and the regression line of y on x was drawn, as shown [2] below.(a) How many different 4-digit numbers can be made? (b) How many different odd 4-digit numbers can be made? y

[3]

(ii) Andrea’s 100 4 discs are put back in the bag. Martin then chooses 4 discs at random, without replacement. Find the probability that (a) the 4 digits include at least 3 odd digits,

[4]

80 (b) the 4 digits add up to 28.

[3] (Q9, Jan 2012)

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physicsandmathstutor.com 4

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(i) 5 of the 7 letters A, B, C, D, E, F, G are arranged in a random order in a straight line. 800 arrangements 1000 of 5 letters 1200 are possible? 1400 (a) How many different



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[2]

(b) actual The list of 7ofpeople includes Jill and Jo. The A group people is(xchosen random fromincluded the list. (iv) The value the IMR for Tanzania is 96. data of for5Tanzania = 1300,aty = 96) is now Given that either Jill andCalculate Jo are both of them is chosen, find the probabilityr,that with the original 6 countries. thechosen value or of neither the product moment correlation coefficient, for both of them are chosen. [3] all 7 countries. [4] (Q7, June 2012) (v) The IMR is now redefined as the infant mortality rate per hundred instead of per thousand, and the (i) value The random X hasfor thealldistribution B(30, 0.6). calculation Find P(X !state 16). what effect, if any, this would [2] of r is variable recalculated 7 countries. Without have on the value of r found in part (iv). [1] (ii) The random variable Y has the distribution B(4, 0.7).

(a) Find = 2). 3-digit numbers can be formed using the digits 1, 2 and 3 when (i) How manyP(Y different (b) no Three values of are chosen at random. Find the probability that their total is 10. (a) repetitions areY allowed,

© OCR 2012



x

(b) GDP How for many of these arrangements endiswith a vowel (AUse or E)? (ii) The another country, Tanzania, 1300 dollars. the regression line in the diagram [3] to estimate the IMR of Tanzania. [1] (ii) A group of 5 people is to be chosen from a list of 7 people. (iii) The GDP for Nigeria is 2400 dollars. Give two reasons why the regression line is unlikely to give a (a) How many different groups of 5 people can be chosen? [1] reliable estimate for the IMR for Nigeria. [2]

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[2] [6] [1]

4732Jan12

(b) any repetitions are allowed, [2] (i) A clock is designed to chime once each hour, on the hour. The clock has a fault so that each time it is 1 the supposed chime is a constant probability (c) each to digit maythere be included at most twice? of 10 that it will not chime. It may be assumed that [2] clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day. Find4-digit the probability that be theformed first time it does not chime is 3 when each digit may be (ii) How many different numbers can using the digits 1, 2 and included at most twice? [5] (a) at 0600 on that day, [3] (Q4, Jan 2013)

(b) before 0600 on that day.

[3]

(ii) Another clock is designed to chime twice4732/01 eachJan13 hour: on the hour and at 30 minutes past theTurn over hour. This 1 clock has a fault so that each time it is supposed to chime there is a constant probability of 20 that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day.

© OCR 2013

(a) Find the probability that the first time it does not chime is at either 0030 or 0130 on that day. [2]