Two or more different variables

5 Each drink from a coffee machine contains \(X \mathrm {~cm} ^ { 3 }\) of coffee and \(Y \mathrm {~cm} ^ { 3 }\) of milk, where \(X\) and \(Y\) are independent variables with \(X \sim \mathrm {~N} \left( 184,15 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 50,8 ^ { 2 } \right)\). If the total volume of the drink is less than \(200 \mathrm {~cm} ^ { 3 }\) the customer receives the drink without charge.

1. Find the percentage of drinks which customers receive without charge.
2. Find the probability that, in a randomly chosen drink, the volume of coffee is more than 4 times the volume of milk.

5 Fiona and Jhoti each take one shower per day. The times, in minutes, taken by Fiona and Jhoti to take a shower are represented by the independent variables \(F \sim \mathrm {~N} \left( 12.2,2.8 ^ { 2 } \right)\) and \(J \sim \mathrm {~N} \left( 11.8,2.6 ^ { 2 } \right)\) respectively. Find the probability that, on a randomly chosen day,

1. the total time taken to shower by Fiona and Jhoti is less than 30 minutes,
2. Fiona takes at least twice as long as Jhoti to take a shower.

2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).

8 In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 57,13 )\) and \(Y \sim \mathrm {~N} ( 28,5 )\). You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating \(X + 2.5 Y\). A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate

1. has a final score that is greater than 140 ,
2. obtains at least 20 more marks in the theory paper than in the practical paper.

4 Each year a transport firm uses \(X\) litres of gasoline and \(Y\) litres of diesel fuel, where \(X\) and \(Y\) have the independent distributions \(X \sim \mathrm {~N} ( 10700,950 ) ^ { 2 }\) and \(Y \sim \mathrm {~N} \left( 13400,1210 ^ { 2 } \right)\).

1. Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.
   The costs per litre of gasoline and diesel fuel are \\(0.80 and \\)0.85 respectively.
2. Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between \(\\) 20000\( and \)\\( 22000\).

7 Before a certain type of book is published it is checked for errors, which are then corrected. For costing purposes each error is classified as either minor or major. The numbers of minor and major errors in a book are modelled by the independent distributions \(\mathrm { N } ( 380,140 )\) and \(\mathrm { N } ( 210,80 )\) respectively. You should assume that no continuity corrections are needed when using these models. A book of this type is chosen at random.

1. Find the probability that the number of minor errors is at least 200 more than the number of major errors.
   The costs of correcting a minor error and a major error are 20 cents and 50 cents respectively.
2. Find the probability that the total cost of correcting the errors in the book is less than \(\\) 190$.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 9.2,12.1 )\) and \(\mathrm { N } ( 3.0,8.6 )\) respectively. Find \(\mathrm { P } ( X > 3 Y )\).

3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.

4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled by three independent random variables \(P , Q\) and \(R\), where \(P \sim \mathrm {~N} \left( 46,19 ^ { 2 } \right) , Q \sim \mathrm {~N} \left( 53,23 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 25,10 ^ { 2 } \right)\). The total value of the minerals found in 1 tonne of rock is modelled by the random variable \(V\), where \(V = P + Q + 2 R\). Use the model to find the probability of finding minerals with a value of at least 93 in a randomly chosen tonne of rock.

2 A factory manufactures paperweights consisting of glass mounted on a wooden base. The volume of glass, in \(\mathrm { cm } ^ { 3 }\), in a paperweight has a Normal distribution with mean 56.5 and standard deviation 2.9. The volume of wood, in \(\mathrm { cm } ^ { 3 }\), also has a Normal distribution with mean 38.4 and standard deviation 1.1. These volumes are independent of each other. For the purpose of quality control, paperweights for testing are chosen at random from the factory's output.

1. Find the probability that the volume of glass in a randomly chosen paperweight is less than \(60 \mathrm {~cm} ^ { 3 }\).
2. Find the probability that the total volume of a randomly chosen paperweight is more than \(100 \mathrm {~cm} ^ { 3 }\). The glass has a mass of 3.1 grams per \(\mathrm { cm } ^ { 3 }\) and the wood has a mass of 0.8 grams per \(\mathrm { cm } ^ { 3 }\).
3. Find the probability that the total mass of a randomly chosen paperweight is between 200 and 220 grams.
4. The factory manager introduces some modifications intended to reduce the mean mass of the paperweights to 200 grams or less. The variance is also affected but not the Normality. Subsequently, for a random sample of 10 paperweights, the sample mean mass is 205.6 grams and the sample standard deviation is 8.51 grams. Is there evidence, at the \(5 \%\) level of significance, that the intended reduction of the mean mass has not been achieved?

2 In a particular chain of supermarkets, one brand of pasta shapes is sold in small packets and large packets. Small packets have a mean weight of 505 g and a standard deviation of 11 g . Large packets have a mean weight of 1005 g and a standard deviation of 17 g . It is assumed that the weights of packets are Normally distributed and are independent of each other.

1. Find the probability that a randomly chosen large packet weighs between 995 g and 1020 g .
2. Find the probability that the weights of two randomly chosen small packets differ by less than 25 g .
3. Find the probability that the total weight of two randomly chosen small packets exceeds the weight of a randomly chosen large packet.
4. Find the probability that the weight of one randomly chosen small packet exceeds half the weight of a randomly chosen large packet by at least 5 g .
5. A different brand of pasta shapes is sold in packets of which the weights are assumed to be Normally distributed with standard deviation 14 g . A random sample of 20 packets of this pasta is found to have a mean weight of 246 g . Find a \(95 \%\) confidence interval for the population mean weight of these packets.

1 Andy, a carpenter, constructs wooden shelf units for storing CDs. The wood used for the shelves has a thickness which is Normally distributed with mean 14 mm and standard deviation 0.55 mm . Andy works to a design which allows a gap of 145 mm between the shelves, but past experience has shown that the gap is Normally distributed with mean 144 mm and standard deviation 0.9 mm . Dimensions of shelves and gaps are assumed to be independent of each other.

1. Find the probability that a randomly chosen gap is less than 145 mm .
2. Find the probability that the combined height of a gap and a shelf is more than 160 mm . A complete unit has 7 shelves and 6 gaps.
3. Find the probability that the overall height of a unit lies between 960 mm and 965 mm . Hence find the probability that at least 3 out of 4 randomly chosen units are between 960 mm and 965 mm high.
4. I buy two randomly chosen CD units made by Andy. The probability that the difference in their heights is less than \(h \mathrm {~mm}\) is 0.95 . Find \(h\).

4 A company that makes meat pies includes a "small" size in its product range. These pies consist of a pastry case and meat filling, the weights of which are independent of each other. The weight of the pastry case, \(C\), is Normally distributed with mean 96 g and variance \(21 \mathrm {~g} ^ { 2 }\). The weight of the meat filling, \(M\), is Normally distributed with mean 57 g and variance \(14 \mathrm {~g} ^ { 2 }\).

1. Find the probability that, in a randomly chosen pie, the weight of the pastry case is between 90 and 100 g .
2. The wrappers on the pies state that the weight is 145 g . Find the proportion of pies that are underweight.
3. The pies are sold in packs of 4 . Find the value of \(w\) such that, in \(95 \%\) of packs, the total weight of the 4 pies in a randomly chosen pack exceeds \(w \mathrm {~g}\).
4. It is required that the weight of the meat filling in a pie should be at least \(35 \%\) of the total weight. Show that this means that \(0.65 M - 0.35 C \geqslant 0\). Hence find the probability that, in a randomly chosen pie, this requirement is met.

1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.

Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)

2. A workshop makes two types of electrical resistor. The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm {~N} ( 20,4 )\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm {~N} ( 10,0.84 )\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent. Find

1. \(\mathrm { E } ( R )\),
2. \(\operatorname { Var } ( R )\),
3. \(\mathrm { P } ( 28.9 < R < 32.64 )\) (6)

3. Given the random variables \(X \sim \mathrm {~N} ( 20,5 )\) and \(Y \sim \mathrm {~N} ( 10,4 )\) where \(X\) and \(Y\) are independent, find

1. \(\mathrm { E } ( X - Y )\),
2. \(\operatorname { Var } ( X - Y )\),
3. \(\mathrm { P } ( 13 < X - Y < 16 )\).

7. The random variable \(D\) is defined as $$D = A - 3 B + 4 C$$ where \(A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)\) and \(C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)\), and \(A , B\) and \(C\) are independent.

1. Find \(\mathrm { P } ( \mathrm { D } < 44 )\). The random variables \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) are independent and each has the same distribution as \(B\). The random variable \(X\) is defined as $$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C .$$
2. Find \(\mathrm { P } ( X > 0 )\). \section*{END}

1. The random variable \(A\) is defined as

$$A = B + 4 C - 3 D$$ where \(B\), \(C\) and \(D\) are independent random variables with $$B \sim \mathrm {~N} \left( 6,2 ^ { 2 } \right) \quad C \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right) \quad D \sim \mathrm {~N} \left( 4,1.5 ^ { 2 } \right)$$ Find \(\mathrm { P } ( A < 45 )\)

3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes. Find the probability that in a randomly chosen week

1. the pupil spends more than 2 hours in total doing French and English homework,
2. the pupil spends more than twice as long doing English homework as he spends doing French homework.
   (6 marks)

1 The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} ( \mu , 11 ) , \quad Y \sim \mathrm {~N} \left( 10 , \sigma ^ { 2 } \right) \quad \text { and } \quad 2 X - 5 Y \sim \mathrm {~N} ( 0,144 ) .$$ Find

1. the values of \(\mu\) and \(\sigma ^ { 2 }\),
2. \(\mathrm { P } ( X - Y > 10 )\).

7 The total mass of a can of pears is the sum of three independent random variables: the mass of pears, the mass of juice, and the mass of the container. The mass in grams of pears in a can has the distribution \(\mathrm { N } ( 300,400 )\). The mass in grams of juice has the distribution \(\mathrm { N } ( 200,60 )\). The mass in grams of the container has the distribution \(\mathrm { N } ( 70,10 )\).

1. Find the probability that the total mass of a randomly chosen can is less than 530 g .
2. Find the probability that the mass of the container of a randomly chosen can is more than one eighth of the total mass of the can.

7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.

The random variable \(R\) is defined as \(R = X + 4Y\) where \(X \sim \text{N}(8, 2^2)\), \(Y \sim \text{N}(14, 3^2)\) and \(X\) and \(Y\) are independent. Find

1. E\((R)\), [2]
2. Var\((R)\), [3]
3. P\((R < 41)\) [3]

The random variables \(Y_1\), \(Y_2\) and \(Y_3\) are independent and each has the same distribution as \(Y\). The random variable \(S\) is defined as $$S = \sum_{i=1}^{3} Y_i - \frac{1}{2}X.$$

1. Find Var\((S)\). [4]

The three tasks most frequently carried out in a garage are \(A\), \(B\) and \(C\). For each of the tasks the times, in minutes, taken by the garage mechanics are assumed to be normally distributed with means and standard deviations given in the following table. Assuming that the times for the three tasks are independent, calculate the probability that

1. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes, [5]
2. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\). [5]

A machine is selecting independently and at random long rods and short rods. The length of the long rods, \(X\) cm, is normally distributed with mean 25 cm and variance 3 cm\(^2\) and the length of the short rods, \(Y\) cm, is normally distributed with mean 15 cm and variance 2 cm\(^2\). Assume that \(X\) and \(Y\) are independent random variables.

1. One long rod and one short rod are chosen at random. Find the probability that the difference in the lengths, \(X - Y\), is between 8 cm and 11 cm. [4]
2. Two long rods and two short rods are chosen at random and are assembled into an approximately rectangular frame. Find the probability that the perimeter of the resulting frame is more than 75 cm. [4]