Revision exercises (Chapters 1 to 6)

197 Revision exercises (Chapters 1 to 6) 1 A car sales company offers buyers a choice, with respect to a particular model, of four colours, three engines and two kinds of transmission. a How many distinguishable cars of the model can the company offer? b If customers can further choose to have or not have each of three additional features, how many distinguishable cars are now possible? 2 In how many different ways can a committee of seven people be selected from eight men and five women if each committee consists of at least four men and at least two women? 3 Xis a continuous random variable defined by: f(x) =,,. e 2-v27r Find: a Pr(X 0) b Pr(X 0 I X 2). 1 --W; 2y 4 A carton contains eight cubes, five of which are white and the remainder are black. A random sample of three cubes is drawn, without replacement, from the carton. Find the probability-that the three cubes are the same colour. 5 If the probability that event B happens is twice the probability that event A happens, and Pr(A U B) = 0.75, find Pr(A) in each of the following cases: a A and B are mutually exclusive b A CB c A and B are independent. 6 Let a random variable, X, have probability distribution: Find the mean and variance of X. X 0 Pr(X = x) 0.5 0.3 7 A certain population of plants has a distribution of heights, measured in centimetres, which is normal with mean 30 cm and standard deviation 2 cm. a Calculate the probability that a randomly selected plant will be less than 27 cm in height. b. If five plants are selected at random, what is the probability that, at most, one is less than 27 cm in height? c If 50 plants are selected at random, estimate the probability that, at most, one is less than 27 cm in height, using: (i) the normal approximation (ii) the Poisson approximation to the binomial distribution. 8 The random variable, Y, is defined by: Find Pr(Y 1) Pr(y = y) = e-o.s (0.5)Y 0 1 2 3 I ' y = ' ' '... y. 9 If Xis a Poisson random variable and Pr(X = 0) = 0.1, find the mean of X.

198 REVISION EXERCISES /) / 10 Packages, each containing ten manufactured articles, are subjected to the following sampling plan. Three articl_es, selected at ranclom from a_package, are tested and the package is accepted if no defective articles are shown up bytheiest. Consider packages which contain exactly oiiedefectiye:art:ide. a If one such package is selected at random, find the probability that exactly one of them will be accepted. b If five such packages are selected at random, find the probability that exactly one of them will be accepted. 11 Two football teams, Richwood and Hawthendon, are to meet in the grand final. Hawthendon prefers to play in fine weather and, for a fine day, Pr [Hawthendon wins] = 0.7. For a wet day, however, Pr [Hawthendon wins] = 0.4. If the probability that the day of the grand final is wet is ' find the probability that Hawthendon wins. 12 If, for two events A and B, Pr(A) = 0.6, Pr(A U B) = 0.8, and Pr(A I B) = 0.6, find Pr(B). 13 A journey of 240 km is to be made using one of two available vehicles, A and B. Vehicle A averages 80 km/h, while vehicle B averages only 60 km/h. The probability that vehicle A is used for the journey is½, and the probability that Bis used isl Find the mean and standard deviation of the time taken for the journey. 14 Determine how many four-letter 'codes' can be formed from the six letters ABCDEF if: a letters can be repeated. b no letters are repeated c each code must contain the letter 'A', and no letters are repeated. 15 In a set of tennis, A is leading 5 games to 3. What is A's chance of winning the set 7 games to 5, if the chance of winning any one game is? 16 The following table gives tl)_e frequency distribution of the number, X, of demands per week for a certain item in a retail store over a period of 100 weeks. I,: I I I,: I,: I : I X 5 Frequency 29 2 : >o a Find the mean and variance of X. b At the beginning of each week the retailer has three items in supply. Assuming that the number of demands per week is a Poisson variable, on how many weeks should the demand exceed the supply over the 100-week period? 17 Three red, two white, and four blue balls are to be placed in a row. How many distinguishable arrangements are possible? 18 If Xis a normally distributed random variable with mean 2 and variance 4, find: a Pr(X < -2) b the value of a for which Pr(X> a) = 0.90. I

REVISION EXERCISES 199 19 A campany produces electronic components at each of two factories, one in Fitzroy, the other in Balwyn. Of the components produced at Fitzroy, 5% are defective while, of those produced at Balwyn, 8% are defective. If 40% of the components are produced r at Fitzroy, find: a the overall proportion of defective components produced b the probability that a component selected at random from the company's production and found to be defective was produced at the Balwyn factory. 20 Let X be the number of heads in four spins of a biassed coin. If the distribution of X has mean 1, find the probability that Xis 0. 21 A gardener plants ten seeds. The probability that a seed will germinate is 0. 9. What is the probability that at least nine of them will germinate? 22 Show that for any events A and B: Pr(A) = Pr(B) Pr(A I B) + Pr(B') Pr(A I B') where B' is the event complement ry to B. 23 A box contains five balls, three of which are white and two of which are black. Two balls are selected at random without replacement. If X denotes the number of black balls selected, find Pr(X = x), x = 0, 1 and 2. Find the mean of X. 24 The following table gives the frequency distribution over one year of the number of accidents per week at an intersection. Number of accidents per week Frequency 0 19 1 20 2 9 3 3 4 0 5 1 more than 5 0 Total= 52 a State the form of the most appropriate theoretical probability distribution for the number of accidents per week. b If the mean of such a distribution is r., give its standard deviation in terms of r.. c Calculate the mean and standard deviation of the frequency distribution given above. State whether the values you obtain support your answer to Question 24a, giving reasons. d Calculate an estimate of the probability that, in a randomly selected week, there is no accident at the intersection: (i) using only the given frequency table (ii) using the theoretical distribution proposed in Question 24a. 25 The loads, Xtonnes, on non-overlapping sections in a warehouse are assumed to be independent with the same normal distribution. Find the mean and standard deviation ofxifpr(x>4) = 0.12 andpr(x<2) = 0.12. 26 A survey was made of 235 houses to determine what type of heating was used - gas, electricity, or oil. It was found that all of these houses used at least one of these forms. Of the 20 homes using oil, eight also used gas, 10 also used electricity, and two also used both s and electricity. 175 houses were either all-gas or all-electricity and 132 used electritity for some heating. If one of these houses is selected atrarulofil,what is the probability that it has a gas connection? "-. (\ \

200 REVISION EXERCISES 27 A packaging machine dispenses nominal half-kilogram lots of a powder. In practice, the weights of individual lots are found to vary randomly, the frequency distribution below being a summary of the weights of 100 lots. Using these data, calculate an estimate, m, of the mean and an estimate, s, of the standard deviation of the weights. Round your results to one decimal place. Weight interval Frequency (grams) (number of lots) 493-496 1 496-499 7 499-502 11 502-505 18 505-508 24 508-511 20 511-514 14 514-517 3 517-520 2 Assuming that the distribution of weights is normal, and that its mean and standard deviation may be taken as m ands respectively, find the proportion of lots whose weights are less than the nominal value. 28 A random sample of 200 patients was treated with a certain drug and 160 were cured. Give 2 sigma confidence limits for the overall number of cures and, therefore, the proportion of cures. 9 A box contains five black and 10 white balls. Find the probability that two balls drawn at random are of different colours if: a the first ball is replaced before the second is drawn b the balls are chosen without replacement. 30 a Compute the sample mean and sample standard deviation of the following set of observations, which have been drawn from a population having a normal distribution: 1.8\ l'.9 2.4 '2.7 2.9 2:9 3.0 3:1 3.3 3.4 3.5 3.9. b Assuming that the true mean and standard deviation of the distribution are equal to the sample values, calculate the probability that an observation exceeds the mean, given that it is less than 3.5. 31 In a rowing eight the seating positions are numbered from 1 to 8, odd numbers denoting 'bow-side' rowers and even numbers 'stroke-side' rowers. If, in a squad of eight rowers, four can row only stroke-side and four can row only bow-side, how many different seating arrangements are possible? \/ 32 a A manufacturer produces rectangular metal sheets, of dimensions 5 cm by 10 cm, which are subject to flaws occurring at the rate of 0.01 flaws per square centimetre. Assuming that the number of flaws per sheet follows a Poisson distribution, find the probability that a sheet contains: (i) no flaws (ii) exactly one flaw (iii) more than one flaw. b If sheets with no flaws realise a profit of $3.00, sheets with one flaw a profit of $1.00, and sheets with more than one flaw result in a loss of $2.00, find the expected or mean profit made per sheet produced. 33 The diameters of pistons produced by a machine are known to be normally distributed. Only those pistons whose diameters lie between 4.993 cm and 6.007 cm are acceptable.

REVISION EXERCISES 20,1 a Find the mean and standard deviation of the distribution of diameters if four per cent of pistons are rejected as being too small, and four per cent are rejected as being too large. b Determine the probability that fewer than two pistons are rejected in a batch of IO such pistons. 34 As a result of a random experiment, E, the events A and B may occur. The events A and Bare independent and Pr(A) = 0.5, Pr(B) = 0.2. a Find the probability that both A and B may occur. Then find the probability that neither A nor B occurs. b Let Z denote the random variable which counts the number of the events A and B which occur as a result of the random experiment E. For example, Z = 0 if neither A nor B occurs. Find Pr(Z = z) for z = 0, 1 and 2, and, therefore, find the mean and variance of Z. c Suppose that the random experiment Eis repeated independently five times. Find the probability that B occurs exactly once. 35 a Screws arrive at an assembly plant in large batches. These batches are of two types: type I contains 50Jo defective screws, and type II contains IOOJo defective screws. On average, 90% of the batches are of type I, and 1 OOJo are of type II. Find the average percentage of defective screws arriving at the assembly plant. b If a batch is selected at random and two screws are selected at random from the batch, find: (i) the probability that the two screws selected are defective (ii) the conditional probability that the batch is of type I, given that the two screws selected are defective. 36 A batch of 1000 light bulbs contains 200 which are defective. A random sample of 100 light bulbs is selected from this batch. Find the mean and the standard deviation of X, the number of defective light bulbs in the sample. Specify an interval within which X will be with a probability of about 0.95. 37 The resistance of an electronic component is required to lie in the range 50 ± 5 ohm; if this condition is not satisfied the component is rejected as unsatisfactory. The components are mass produced, and the resistance of a randomly selected component is normally distributed with mean 51 ohm and standard deviation 4 ohm. a Find the probability that a randomly selected component is unsatisfactory. b If satisfactory components yield a profit of 20 cents while unsatisfactory components must be scrapped at a loss of 10 cents, find the mean profit per component. 38 If two fair dice are rolled repeatedly, find the probability that a double (i.e. two numbers the same) is obtained before an odd sum is obtained. 39 Suppose that smokers are ten times as likely as non-smokers to develop lung cancer. If 40% of the adult population aresmokers, and 1 OJo of the population develop lung cancer, find the probability that a smoker develops lung cancer. 40 In a certain population, the number of genetic mutations which occur over any given period of time has a Poisson distribution with, on average, 1.2 mutations occurring per day. a Find the probability that, in a particular day, the actual number of mutations is above average. b Find the probability that the number of mutations in a particular day differs from the mean by less than one standard deviation. c Find the smallest number of hours, such that the probability of having at least one mutation during these hours is at least 0.9.

202 REVISION EXERCISES 41 A bag contains six balls, each of which is known to be either purple or green. A random sample of four balls is drawn from the bag, without replacement, and is found to contain three purple balls and one green one. Find the mixture of balls in the bag which gives the highest probability of this sample being drawn, and evaluate this probability. 42 An alarm clock, used to wake a person for work, has a probability of 0.9 of ringing in the morning. If the alarm rings, there is a probability of 0.8 that the person gets to work on time; but, if the alarm does not ring, the probability of getting to work on time is only 0.4. Find: a the overall probability of arriving at work on time b the probability that, on a randomly chosen morning among those on which the person was late, the alarm did ring. (43 In a certain community the probability of a child being female is 0.5 5. Find the probability that a couple with exactly four children has an odd number of girls. 44 Xis a geometric random variable such that: Find Pr(X 3). Pr(X = x) = (0.1) (0.9) X, x = 0, l, 2,... 45 An unbiassed die is to be thrown 180 times. Using the normal approximation for the binomial distribution, find the probability that the 6 will show on the uppermost face exactly 31 times. 46 The life of a certain make of car battery is known to be normally distributed with mean 30 months and standard deviation 6 months. The batteries cost the manufacturer $20 to make, and they are sold for $30. If half the purchase price is refunded on any battery which does not last 24 months, what average profit will the manufacturer make per battery? ( 47 Three fair dice with faces numbered one to six are rolled. Find the probability that the three numbers showing uppermost are: a the same b all different. 48 A chain is being made by joining successive links. Each link has a probability of of being 'strong' and a probability of½ of being 'weak', independently of other links. a What is the probability that the first n consecutive links will all be strong? b What is the probability that there will be n strong links followed by the first weak_ link? c What is the largest value n can have so that a chain of n links consists of all strong links with a probability of at least½? 49 A box contains ten transistors, of which four are defective. A sample of three is taken from the box, without replacement. Let Xbe the number of defective transistors in the sample. a Find Pr(X = 1). b State the mean of X. 50 If Pr(A) = 0.40, Pr(B) = 0.70 and Pr(A U B) = 0.8 5, find: a Pr(A '), where A' is the event complementary to A b Pr(A n B) c Pr(A' U B') d Pr(A I (A U B)). Determine whether events A and Bare independent, giving a reason.

REVISION EXERCISES 203 51 When three targetshooters take part in a shooting contest, their respective chances of hitting the target are½, ½and¼, Calculate the probability that exactly one bullet will hit the target if all shooters fire at it simultaneously. 52 Consider a random variable X with probability distribution given by: X Pr(X = x) Find, approximately, the probability that no more than ten zeros are obtained in twenty-four independent observations of X. 53 Suppose that the strengths of mass-produced items are normally distributed with mean µ and standard deviation 0.5. The value ofµ can be controlled by a machine setting. If the strength of an item is less than 5, it is classified as defective. Revenue from sales of non-defective items is $R per item, while revenue from defective items is $0. lr per item. The cost of production of items with mean µ is $0. lrµ per item. a Find the expected profit per item if µ = 6. b Indicate on a rough sketch the behaviour of the expected profit per item as a function ofµ. 54 Consider a sequence of independent trials each having probability of success p. Let X denote the number of successes in the first ten trials, and let Y denote the number of failures before the first success. a Specify the range of values of p for which: Pr(X = 0) Pr(X = 1). b Specify the range of values of p for which: Pr(Y = 0) Pr(Y = 1). 55 A box contains five red and five blue balls. Find the probability of obtaining two red and two blue balls in a sample of four: a if the sample is drawn with replacement b if the sample is drawn without replacement. 56 The variables X, Yand Z are defined as follows: e-3 3x Pr(X = x) = x = 0, 1, 2, 3,... Pr(Y = y)= (0.4) Y 0.6,y = 0, 1, 2, 3,... Pr(Z = z) = ( 0 ) (0.4) z (0.6) 10 -z, z = 0, 1, 2,... 10 a Write the values of the meanµ for each of the variables X, Y and Z. b Find Pr(X = 3), Pr(Y 1), Pr(Z 2). 57 For each of the following random variables, state the probability distribution which is most likely to apply: a the number of times per month that a machine breaks down in a factory b the number of attempts a student makes at a mathematical problem before finally solving it q\ dzfl--c \c,1(,, c the number 'M students wearing glasses in a group of twenty students,, ' d the number of accidents per week on a country road e the diameters of mass-produced metal rods f the measurements of the length of a football field

204 REVISION EXERCISES 58 a Five tiles are laid end to end. If three tiles are black and the other two are white, how many different patterns are possible? b Three red, two white and four blue balls are to be placed in a row. How many distinguishable arrangements are possible? 59 A die is tossed 240 times. On how many of these occasions would we expect an odd number to turn up? Give limits between which the number of occasions will lie with a probability of about 0.95. 60 a How many different arrangements can be made with the letters of the word 'requisition'? If they are arranged at random, what is the probability that q is immediately followed by u? b In how many ways can a committee of 10 be chosen from six English people, seven French people and three Australians, if each committee contains at least five English people and at least two Australians? 61 A manufacturer finds that 10 per cent of the articles made in the factory are defective. If three articles are taken at random, what is the probability that: a all are defective? b none is defective? c there are more defectives than non-defectives? 62 In one country, 100 men from a random sample of 400 were more than 180 cm tall and, in a nearby country, the number was 300 from a random sample of 800. Does this indicate that there is a greater proportion of men above this height in one country than in the other?