1. The masses, x grams, of the contents of 25 tins of Brand A anchovies are summarized by x =

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1 P6.C1_C2.E1.Representation of Data and Probability 1. The masses, x grams, of the contents of 25 tins of Brand A anchovies are summarized by x = and x 2 = Find the mean and variance of the masses. What is the unit of measurement of mean and the variance? [5] 2. The mass of each of 60 pebbles collected from Muara beach was measured. The results, correct to the nearest gram, are summarized in the following table: Mass Frequency (i) Estimate the mean and standard deviation of this data. (ii)draw a histogram of the data. Draw the cumulative frequency curve and use it to estimate (iii) the median and interquartile range. (iv) draw a box and whisker plot. 3. Out of 100 applicants for the post of the general manager of a company, 80 are male. 70% of the male applicants hold MBA degrees while 90% of the female applicants hold MBA degrees. An applicant is chosen at random, find the probability that (a) the applicant is a female who does not hold an MBA degree, (b) the applicant holds an MBA degree. Hence, or otherwise, find the conditional probability that the applicant who does not hold an MBA degree is a male. 4. A bag contains 8 pens, of which 3 are red and 5 are blue. A pen is picked at random from the bag and its colour is noted. The pen is not replaced. A second pen is then picked. Find the probability that (a) the first pen is blue, (b) the first is blue and the second is red, (c) a red pen is picked, given that the first pen is blue. [32 marks]

2 P6.C1_C2.E2.Representation of Data and Probability 1. The speeds, in km/h, of 200 vehicles traveling on a motorway were measured by a radar device. The results are summarized in the following table. Speed Frequency (i) Estimate the mean and standard deviation of this data. (ii)draw a histogram of the data. 2. The following data are the heights of 39 people in cm (correct to the nearest cm), taken from the data fill ʻBrain Sizeʼ (a) Construct a stem-and-leaf diagram for the data. (b) Find the median and interquartile range. (c) Draw the box plot. 3. If events A and B are such that they are independent and P(A) = 0.6, P(B) = 0.3, find (a) P (A B) (b) P (A B) (c) P (A B) State, giving a reason, whether the events A and B are mutually exclusive Weather record indicate that the probability that a particular day is dry is PTE Katok is a football team 10. whose record of success is better on dry days than on wet days. The probability that PTE Katok wins on 3 a dry day is whereas the probability that they win on a wet day is. PTE Katok is due to play their next match on Saturday. (a) What is the probability that PTE Katok will win? (b) Three Saturdays ago, PTE Katok won their match. What is the probability that it was a dry day? [30 marks]

3 P6.C1_C2.E3.Representation of Data and Probability 1. In a tea shop, 70% of customers order tea with milk, 20% tea with lemon and 10% tea with neither. Of those taking tea with milk 2/5 take sugar, of those taking tea with lemon 1/4 take sugar, and of those taking tea with neither milk nor lemon 11/20 take sugar. A customer is chosen at random. (a) Represent the information given on a tree diagram and use it to find the probability that the customer takes sugar. (b) Find the probability that the customer takes milk or sugar or both. (c) Find the probability that the customer takes sugar and milk. Hence find the probability that the customer takes milk given that the customer takes sugar. 2. A bag contains six red and four green counters. Two counters are drawn, without replacement. Use a carefully labelled tree diagram to find the probabilities that (a) both counters are red, (b) both counters are green, (c) just one counter is red, (d) at least one counter is red, (e) the second counter is red. 3. In a certain part of the world, there are more wet days than dry days. If a given day is wet, the probability that the following day will also be wet is 0.8. If a given day is dry, the probability that the following day will also be dry is 0.6. Given that Wednesday of a particular week is dry, calculate the probability that (a) Thursday and Friday of the same week are both wet days, (b) Friday of the same week is a wet day. In one season there are 44 cricket matches, each played over three consecutive days, in which the first and third days were dry. For how many of these matches would you expect that the second day was wet? 4. The heights, x cm, of a sample of 80 female students are summarized by the equations!!! (x! 160)! = 240 and (x 160) 2 = 8720 Find the mean and standard deviation of the heights of the 80 female students. [7] [33 marks]

4 P6.C1_C2.E4.Representation of Data and Probability 1. The land area, of 80 houses built on an estate were recorded to the nearest 10m 2 and grouped in the table below: Land Area 200 A< A< A< A< A<900 Frequency (i) Show that the mean is 500 and estimate the standard deviation of this data. Draw the cumulative frequency curve and use it to estimate (ii) the median and interquartile range. (iii)draw a box and whisker plot. (iv)the number of houses with a land area of more than 620 m 2. [5] 2. The table shows the average daily traveling times, x minutes, for a sample of 201 people who commute daily to work in Singapore. Time (mins) 30 x<45 45 x<60 60 x<75 75 x<90 90 x<120 Frequency (a) Calculate estimates of the mean and standard deviation of these times. (b) Illustrate the data by means of an accurate histogram drawn on graph paper. (c) Find the median and the interquartile range of these times. 3. Bag A contains 1 red ball and 1 black ball, and bag B contains 2 red balls; all four balls are distinguishable apart from their colour. One ball is chosen at random from A and is transferred to B. One ball is then chosen at random from B and is transferred to A. (a) Draw a tree diagram to illustrate the possibilities for the colours of the balls transferred from A to B and then from B to A. Show the relevant probability on each branch of your tree diagram. (b) Find the probability that, after both transfers, the black ball is in bag A. 4. Maggie takes examinations in Mathematics, Chemistry and Biology. The probability that she passes Mathematics is 0.7 and the corresponding probabilities for Chemistry and Biology are 0.8 and 0.6. Given that her performances in each subject are independent, draw a tree diagram to show the possible outcomes. Find the probability that Maggie (a) fails all three examinations (b) fails just one examination. Given that Maggie fails just one examination, (c) find the probability that she fails Biology. [40 marks]

5 P6.C3.E1.Permutation and Combination 1. The letters of the word CONSTANTINOPLE are written on 14 cards, one on each card. The cards are shuffled and then arranged in a straight line. (i) How many different possible arrangements are there? (ii)how many arrangements are there where the vowels are together? (iii)how many arrangements are there where the ʻTʼs are at each end? 2. Find the number of ways in which the letter of ISOSCELES can be arranged if the two Eʼs are separated. 3. A committee of six is to be formed from nine women and three men. In how many ways can the members be chosen so as to include at least one man? 4. Find how many numbers greater than can be made, using all digits of [18 marks]

6 P6.C3.E2.Permutation and Combination 1. (i) In how many ways can 2 men be chosen from 5 men? (ii) In how many ways can 3 women be chosen from 4 women? (iii)a mixed basketball team is made up of 2 men and 3 women, chosen from 5 men and 4 women. How many different teams can be formed? 2. How many arrangements can be made of the letters in the word TERRITORY? 3. Nine people are going to travel in two taxis. The larger has five seats, and the smaller has four. In how many ways can the party be split up? 4. The word ALGEBRA consists of 7 letters. (i) Find the total number of different arrangements when using all the 7 letters. (ii)find the number of these arrangements which (a) have the two Aʼs together, (b) have no two Aʼs together. [13 marks]

7 P6.C3.E3.Permutation and Combination 1. A chess club has 10 members, of whom 4 are men and 6 are women. A team of 4 members selected to play in a match. Find the number of different ways of selecting the team if: (a) there is no restriction, (b) there must be an equal number of men and women, (c) there are at least 2 men in the team. 2. (a) How many distinct arrangements are there of the letters in the word ABRACADABRA? (b) How many distinct arrangements are there when the vowels are together? 3. From a group of 30 boys and 32 girls, two girls and two boys are to be chosen to represent PTE Katok. How many possible selections are there? [14 marks]

8 P6.C4_C5.E1.Discrete / Binomial / Normal Distribution 1. A discrete random variable X has the following probability distribution. x P(X=x) p 0 p 1 p 2 Given that E(X) = 4.1 and Var(X) = 4.99, find the values of p 0, p 1 and p 2. [8] 2. Each week a security firm transports a large sum of money between two places. The day on which the journey is made is varied at random, and in any week, each of the five days from Monday to Friday is equally likely to be chosen. (i) Calculate the probability that, in a period of 10 weeks, Friday will be chosen at least 3 times. (ii)the event that, in a 4 week period, the same day is chosen on all four occasions is denoted by S. Find the probability of S occurring. 3. A reader of a magazine enters for a competition in the magazine in which the competitors have to choose the correct answers to a number of questions. There are five suggested answers for each question, but the reader is completely unskilful and selects an answer at random to each question, so that, for each question, the probability of choosing the correct answer is 1/5. (i) For a competition with 12 questions, find the probability of the reader getting more than 3 correct answers (ii) For a competition with 100 questions, use a suitable approximation to estimate the probability of the reader getting more than 26 correct answers, (iii) For a group of 1000 readers, all of whom select answers at random in a competition of 100 questions, find the probability that less than 50 readers will each get more than 6 correct answers. 4. A hockey team consists of 11 players. It may be assumed that, on every occasion, the probability of any one of the regular members of the team being unavailable for selection is 0.15, independently of all the other members. Calculate the probability that on a particular occasion, (i) exactly one regular member is unavailable, (ii)more than two regular members are unavailable. [26 marks]

9 P6.C4_C5.E2.Discrete / Binomial / Normal Distribution 1. Nails are sold in packets of 100. Occasionally a nail is faulty. The number of faulty nails in a randomly chosen packet is denoted by X. Assuming that faulty nails occur independently and at random, calculate the mean and standard deviation of X, given that the probability of any nail being faulty is State the conditions under which a Normal distribution may be used as a good approximation to the Binomial distribution. A die is to be thrown 8 times. Calculate the probability of obtaining no sixes in the 8 throws and the probability of obtaining at least 2 sixes in the 8 throws. Two dice are to be thrown 200 times. Write down an expression but donʼt evaluate an expression for the probability of obtaining two sixes exactly 4 times in the 200 trials. Using normal approximation to a binomial distribution, estimate this probability. 3. A school student investigated how long he actually had to spend on homework assignments, which were nominally for half-hour periods. He found that the times were approximately normally distributed, with mean 35 minutes and standard deviation 8 minutes. Using this model and assuming independence between assignments, find (i) the probability that one particular assignment will take less than 25 minutes. (ii)the time in which, 90% of all the assignments can be completed. (iii)the probability that three assignments each take more than 40 minutes. 4. The discrete random variable X takes the values 1, 2, 3, 4 and 5 only, with the probabilities shown in the table. x P(X=x) a b (a) Given that E(X) = 2.34, show that a = 0.34, and find the value of b. (b) Find Var(X). [28 marks]

10 P6.C4_C5.E3. Discrete / Binomial / Normal Distribution 1. The six faces of a fair cubical die are numbered 1,2, 2, 3, 3 and 3. When the die is thrown once, the score is the number appearing on the top face. This is denoted by X. (a) Find the mean and standard deviation of X. (b) The dice is thrown twice and Y denotes the sum of the scores obtained. Find the probability distribution of Y. Hence find E(Y) and Var(Y). [5] [5] 2. In a certain examination paper there are 10 questions. Each question has 5 suggested answers and the candidates have to choose the right one in each question. Suppose that candidate X chooses answers entirely at random, so that he is equally likely to choose any one of the 5 answers in each question. Calculate the probability that he will score at least 3 correct answers out of 10. In a similar examination, there are 100 questions, each with 5 suggested answers. X again chooses entirely at random. By using a normal approximation to a binomial distribution, estimate the probability of X scoring at least 30 correct answers out of In a certain examination 35% of all candidates pass. Calculate the expectation and variance of the number of passes in a group of 30 randomly chosen candidates who take the examination. 4. The number of times a certain factory machine breaks down each working week has been recorded over a long period. From these data, the following probability distribution for the number, X, of weekly breakdowns was produced. x P(X=x) (a) Find the mean and standard deviation of X. (b) What would be the expected total number of breakdown that will occur over the next 48 working weeks? [22 marks]

11 P6.C4_C5.E4. Discrete / Binomial / Normal Distribution 1. The discrete random variable X has the following probability distribution. x P(X=x) 0.3 a b 0.25 (i) Write down an equation satisfied by a and b. (ii)given that E(X) = 4, find a and b. 2. A box contains 10 pens of which 3 are new. A random sample of two pens is taken. (i) Show that the probability of getting exactly one new pen in the sample is 7/15. (ii)construct a probability distribution table for the number of new pens in the sample. (iii)calculate the expected number of new pens in the sample. 3. A fair cubical die with faces numbered 1, 1, 1, 2, 3, 4 is thrown and the score noted. The area A of a square of side equal to the score is calculated, so, for example, when the score on the die is 3, the value of A is 9. (i) Draw up a table to show the probability distribution of A. (ii)find E(A) and Var(A). 4. Of the customers visiting the stereo section of a large electronics store 75% on average make a purchase. (i) Calculate the probability that, out of 14 customers, at least 12 customers make a purchase. (ii)find the least possible number of customers, given that the probability of all the customers make a purchase is less than 5%. [24 marks]

12 P6.C4_C5.E5. Discrete / Binomial / Normal Distribution 1. X is a normal variable with mean µ and standard deviation σ. It is given that P (X >128) = 0.15 and that P (X >97) = Calculate µ and σ. [5] 2. A bag contains 7 orange balls and 3 blue balls. 4 balls are selected at random from the bag, without replacement. Let X denote the number of blue balls selected. (i) Show that P(X=0) = 1/6 and P(X=1) = 1/2 (ii)construct a table to show the probability distribution of X (iii)find the mean and variance of X. 3. A discrete random variable X takes values 0, 1, 2, 4, 8 with probabilities shown in the table. x P(X=x) p 1/2 1/4 1/8 1/16 (i) Evaluate p. (ii)find E(X) and Var(X). 4. The probability of there being X unusable matches in a full box of Surelite matches is given by P(X=0) = 8k, P(X=1) = 5k, P(X=2) = P(X=3) = k, P(X 4) = 0. Determine the constant k and the expectation and variance of X. [5] [27 marks]

13 P6.C4_C5.E6. Discrete / Binomial / Normal Distribution 1. The random variable X has the following probability distribution. x P(X=x) t 2t 1-3t where 0 <t< 1 3 (i) Find E(X) in terms of t. (ii)find Var(X) in terms of t. 2. A coin and a six-faced die are thrown simultaneously. The random variable X is defined as follows: If the coin shows a head, then X is the score on the die. If the coin shows a tail, then X is twice the score on the die. (i) Find the expected value, (ii)find Var(X). µ of X 3. Some of the eggs sold in a store are packed in boxes of 10. For any egg, the probability that it is cracked 0.05, independently of all other eggs. A shelf contains 80 of these boxes. Calculate the expected value of the number of boxes on the shelf which do not contain a cracked egg. 4. Items from a production line are examined for any defects. The probability that any item will be found to be defective is 0.15, independently of all other items. A batch of 16 items is inspected. Calculate the probability that the number of defective items is (a) exactly 2, (b) at least 3. [21 marks]

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