A firm sells cheese in packets. The mass of cheese in the packets is normally distributed with mean 1.12 kg and standard deviation 0.15 kg. The firm makes a profit of $0.32 per kilogram of cheese sold.

Find the probability that the total profit made on 50 randomly chosen packets of cheese is less than $18.00. [6]

The masses, in grams, of apples of a certain type have the distribution \(\mathrm N(58.4,6.3^2)\). The apples are packed in bags, with each bag containing 10 randomly chosen apples.

Find the probability that a randomly chosen bag contains apples whose total mass is between 550 g and 600 g. [5]

Each week, Birgitte takes three train journeys. The times, in minutes, for these journeys have the independent distributions \[A\sim\mathrm N(28,15),\qquad B\sim\mathrm N(23,8),\qquad C\sim\mathrm N(99,24).\]

For a randomly chosen week, find \(P(C>2(A+B))\). [5]

The volumes of regular and large coffees at a certain café have independent normal distributions. The mean and standard deviation of the volumes of regular coffees are \(215\text{ cm}^3\) and \(2.8\text{ cm}^3\) respectively. The mean and standard deviation of the volumes of large coffees are \(330\text{ cm}^3\) and \(3.3\text{ cm}^3\) respectively. Find the probability that the total volume of three randomly chosen regular coffees is greater than the total volume of two randomly chosen large coffees. [5]

One batch of a cake mixture contains \(X\) litres of ingredient A and \(Y\) litres of ingredient B, where \(X\) and \(Y\) have the independent distributions \(X\sim\mathrm N(25.5,0.07)\) and \(Y\sim\mathrm N(26.0,0.08)\).

(a) Find the probability that, in a randomly chosen batch of the mixture, the number of litres of ingredient A is greater than the number of litres of ingredient B. [5]

The mass of 1 litre of ingredient A is \(m\text{ kg}\) and the mass of 1 litre of ingredient B is \(1.5m\text{ kg}\), where \(m\) is a constant.

(b) Given that the probability that the total mass of a randomly chosen batch of the mixture is less than 70 kg is \(0.3\), find the value of \(m\). [5]

The masses, in kilograms, of large and small bags of potatoes have the independent distributions \(N(2.5,0.05)\) and \(N(0.8,0.02)\) respectively.

(a) Find the probability that the total mass of a randomly chosen large bag of potatoes and a randomly chosen small bag of potatoes is more than 3.55 kg. [5]

(b) Find the probability that the mass of a randomly chosen large bag of potatoes is less than 3 times the mass of a randomly chosen small bag of potatoes. [5]

The random variable \(X\) has a normal distribution with mean 10 and standard deviation 3. The independent random variable \(Y\) has a Poisson distribution with mean 4.

(a) Find the standard deviation of \(X+Y\). [3]

(b) Find the standard deviation of \(5X-Y\). [3]

The masses, in kg, of large and small bags of tomatoes have the distributions \(L\sim N(2.10,0.12)\) and \(S\sim N(1.51,0.09)\) respectively.

(a) Find \(\mathrm P(L>S+0.5)\). [5]

The price of tomatoes is \(\$4.30\) per kg.

(b) A large bag of tomatoes and a small bag of tomatoes are chosen at random. Find the probability that the total price of the tomatoes in the two bags is less than \(\$16\). [5]

At an entertainment centre, the cost for using a particular video game is \(\$0.40\) per minute. The number of minutes for which people use the video game has mean 15 and variance 9.

(a) Find the mean and variance of the amount people pay for using the video game. [3]

Each day, 35 people independently use the video game.

(b) Find the mean and variance of the total amount paid by 35 people. [3]

(a) One of a group of three students is to be chosen at random.

(a) Explain how a single throw of a fair six-sided dice could be used to make the choice. [1]

The times, in minutes, taken by students to complete a test are normally distributed with mean 125 and variance 50. Two students are chosen at random.

(b) Find the probability that the difference between the times taken by these two students to complete the test is more than 12 minutes. [5]

In Urberia, the masses, in kilograms, of men have the distribution \(N(70.3,5.9^2)\). A certain footbridge in Urberia can take a maximum safe load of 1500 kg. When \(n\) men stand on the bridge, the probability that the bridge is unsafe is less than 0.01.

Stating a necessary assumption, find the maximum value of \(n\). [4]

Candidates for a certain diploma take two tests. Their marks for the first test and the second test are modelled by independent variables with distributions \(N(38.1,3.8^2)\) and \(N(64.0,6.1^2)\) respectively. The final mark, \(F\), for each candidate is found by doubling the mark in the first test and adding the result to the mark in the second test.

Find the probability that the mean, \(\bar F\), of the final marks of a random sample of 25 candidates is greater than 143. [5]

The random variables \(X\) and \(Y\) have the independent distributions \(N(44,16)\) and \(N(30,9)\) respectively.

Find \(\mathrm P(X-Y<15)\). [5]

The masses, in kilograms, of small and large bags of wheat have independent distributions \(N(16.0,0.4)\) and \(N(51.0,0.9)\), respectively. Find the probability that the total mass of 3 randomly chosen small bags exceeds the mass of one randomly chosen large bag. [5]

\(X\sim N(31.2,10.4^2)\). Two independent values \(X_1,X_2\) are chosen. Find \(P(X_1>3X_2)\). [5]

\(X\sim N(10,12)\). Independent values \(X_1,X_2\) are chosen.

(a) Write down \(P(X_1>X_2)\). [1]

(b) Find \(P(X_1>2X_2-3)\). [5]

Large and small cement-bag masses have independent distributions \(N(50,2.4)\) and \(N(26,1.8)\) kg. Find the probability that 5 randomly chosen large bags have a greater total mass than 10 randomly chosen small bags. [5]

Yearly gasoline and diesel usage, \(X,Y\) litres, have independent distributions \(N(10700,950^2)\), \(N(13400,1210^2)\).

(a) Find the probability of using more gasoline than diesel in a randomly chosen year. [5]

Gasoline and diesel cost \(\$0.80\) and \(\$0.85\) per litre.

(b) Find the probability that the total annual cost is between \(\$20000\) and \(\$22000\). [5]

The masses, in kilograms, of chemicals A and B produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X\sim N(10.3,5.76)\) and \(Y\sim N(11.4,9.61)\). The income generated by the chemicals is $2.50 per kilogram for A and $3.25 per kilogram for B.

(a) Find the mean and variance of the daily income generated by chemical A. [2]

(b) Find the probability that, on a randomly chosen day, the income generated by chemical A is greater than the income generated by chemical B. [6]

A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(N(0.7,0.02^2)\) and \(N(100.0,3.0^2)\) respectively.

Dried yeast costs $13.50 per kilogram and flour costs $0.90 per kilogram. For making one batch of bread the total of all other costs is $55. The factory sells each batch of bread for $200.

Find the probability that the profit made on one randomly chosen batch of bread is greater than $40. [7]

Large packets of rice are packed in cartons, each containing 20 randomly chosen packets. The masses of these packets are normally distributed with mean 1010 g and standard deviation 3.4 g. The masses of the cartons, when empty, are independently normally distributed with mean 50 g and standard deviation 2.0 g.

(a) Find the variance of the masses of full cartons. [2]

Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean 6730 g and standard deviation 15.0 g. The masses of the boxes and cartons are distributed independently of each other.

(b) Find the probability that the mass of a randomly chosen full carton is more than three times the mass of a randomly chosen full box. [5]

(a) Two random variables \(X\) and \(Y\) have the independent distributions \(N(7,3)\) and \(N(6,2)\) respectively. A random value of each variable is taken. Find the probability that the two values differ by more than 2. [5]

(b) Each candidate’s overall score in a science test is calculated as follows. The mark for theory is denoted by \(T\), the mark for practical is denoted by \(P\), and the overall score is given by \(T+1.5P\). The variables \(T\) and \(P\) are assumed to be independent with distributions \(N(62,158)\) and \(N(42,108)\) respectively. You should assume that no continuity corrections are needed when using these distributions.

(b(i)) A pass is awarded to candidates whose overall score is at least 90. Find the proportion of candidates who pass. [5]

(b(ii)) Comment on the assumption that the variables \(T\) and \(P\) are independent. [1]

The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced each day is independent of other days. The factory makes $50 profit on each tonne of steel produced.

Find the probability that the total profit made in a randomly chosen 7-day week is less than $22 000. [6]

The masses, in grams, of large and small packets of Maxwheat cereal have the independent distributions \(N(410.0,3.6^2)\) and \(N(206.0,3.7^2)\) respectively.

(a) Find the probability that a randomly chosen large packet has a mass that is more than double the mass of a randomly chosen small packet. [5]

The packets are placed in boxes. The boxes are identical in appearance. 60% of the boxes contain exactly 10 randomly chosen large packets. 40% of the boxes contain exactly 20 randomly chosen small packets.

(b) Find the probability that a randomly chosen box contains packets with a total mass of more than 4080 grams. [6]

Each month a company sells \(X\) kg of brown sugar and \(Y\) kg of white sugar, where \(X\) and \(Y\) have the independent distributions \(N(2500,120^2)\) and \(N(3700,130^2)\) respectively.

(a) Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months. [3]

The company makes a profit of \$1.50 per kilogram of brown sugar sold and makes a loss of \$0.20 per kilogram of white sugar sold.

(b) Find the probability that, in a randomly chosen month, the total profit is less than \$3000. [5]

The masses, in grams, of small and large bags of flour have the distributions \(N(510,100)\) and \(N(1015,324)\) respectively. André selects 4 small bags of flour and 2 large bags of flour at random.

(a) Find the probability that the total mass of these 6 bags of flour is less than 4130 g. [5]

(b) Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags. [5]

The lengths, in centimetres, of two types of insect, A and B, are modelled by the random variables \(X\sim N(6.2,0.36)\) and \(Y\sim N(2.4,0.25)\) respectively.

Find the probability that the length of a randomly chosen type A insect is greater than the sum of the lengths of 3 randomly chosen type B insects. [5]

The independent random variables \(X\) and \(Y\) have distributions \(\operatorname{Po}(2)\) and \(B(20,\frac14)\) respectively.

(a) Find the mean and standard deviation of \(X-3Y\). [5]

(b) Find \(P(Y=15X)\). [3]

The masses, in kilograms, of large and small sacks of grain have the distributions \(N(53,11)\) and \(N(14,3)\) respectively.

(a) Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack. [5]

A lift can safely carry a maximum mass of 1000 kg.

(b) Find the probability that the lift can safely carry 12 randomly chosen large sacks and 25 randomly chosen small sacks. [5]

Each box of Seeds & Raisins contains \(S\) grams of seeds and \(R\) grams of raisins. The weight of a box, when empty, is \(B\) grams. \(S\), \(R\) and \(B\) are independent random variables, where \(S\sim N(300,45)\), \(R\sim N(200,25)\) and \(B\sim N(15,4)\). A full box of Seeds & Raisins is chosen at random.

(a) Find the probability that the total weight of the box and its contents is more than 500 grams. [5]

(b) Find the probability that the weight of seeds in the box is less than 1.4 times the weight of raisins in the box. [5]

The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m.

(a) Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m. [4]

(b) Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m. [5]

The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\). A random sample of 40 races run by Tania gave the following results. \[n=40,\qquad\sum t=560,\qquad\sum t^2=7850.\]

(a) Calculate unbiased estimates of \(\mu\) and \(\sigma^2\). [3]

The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(N(14.2,0.3)\).

(b) Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki’s time will be at least 0.1 s more than Tania’s time. [5]

The mass, in kilograms, of a block of cheese sold in a supermarket is denoted by the random variable \(M\). The masses of a random sample of 40 blocks are summarised as follows. \[n=40,\qquad\sum m=20.50,\qquad\sum m^2=10.7280.\]

(a) Calculate unbiased estimates of the population mean and variance of \(M\). [3]

(b) The price, \$\(P\), of a block of cheese of mass \(M\) kg is found using the formula \(P=11M+0.50\).

(b) Find estimates of the population mean and variance of \(P\). [3]

The masses, in kilograms, of large and small sacks of flour have the distributions \(N(55,3^2)\) and \(N(27,2.5^2)\) respectively.

(a) Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg.

(a) Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour. [5]

(b) Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour. [5]

Wendy’s journey to work consists of three parts: walking to the train station, riding on the train and then walking to the office. The times, in minutes, for the three parts of her journey are independent and have the distributions \(N(15.0,1.1^2)\), \(N(32.0,3.5^2)\) and \(N(8.6,1.2^2)\) respectively.

(a) Find the mean and variance of the total time for Wendy’s journey. [2]

If Wendy’s journey takes more than 60 minutes, she is late for work.

(b) Find the probability that, on a randomly chosen day, Wendy will be late for work. [3]

(c) Find the probability that the mean of Wendy’s journey times over 15 randomly chosen days will be less than 54.5 minutes. [3]

The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable \(X\), which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of \$5 to the team for each goal that they score in the next 10 matches.

Find the expectation and standard deviation of the amount that the supporter will pay. [5]

The volumes, in litres, of juice in large and small bottles have the distributions \(N(5.10,0.0102)\) and \(N(2.51,0.0036)\) respectively.

(a) Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres. [5]

(b) Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle. [5]

The masses, in kilograms, of female and male animals of a certain species have the distributions \(N(102,27^2)\) and \(N(170,55^2)\) respectively.

Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. [6]

A book is published in two different versions, \(S\) and \(L\). The number of copies of version \(S\) sold per week has the distribution \(N(380,140)\), and the number of copies of version \(L\) sold per week has the distribution \(N(210,80)\). The numbers of copies of the two versions sold per week are independent.

(a) Find the probability that the number of copies of version S sold in a week is at least 200 more than the number of copies of version L sold in the same week. [5]

The cost of producing each copy of version S is 20 cents, and the cost of producing each copy of version L is 50 cents.

(b) Find the probability that the total cost of producing all the copies of both versions sold in a week is less than \$190. [5]

The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions \(N(40,1.5^2)\) and \(N(12,0.7^2)\) respectively.

(a) Find the probability that the total mass of 6 randomly chosen large sacks of flour is more than 245 kg. [4]

(b) Find the probability that the mass of a randomly chosen large sack of flour is less than 4 times the mass of a randomly chosen small sack of flour. [6]

The masses, in grams, of plums of a certain type have the distribution \(N(40.4,5.2^2)\). The plums are packed in bags, each containing 6 randomly chosen plums. If the total weight of plums in a bag is less than 220 g the bag is rejected.

Find the percentage of bags that are rejected. [4]

Each day at the gym, Sarah completes three runs. The distances, in metres, have the independent distributions \[W\sim N(1520,450),\quad X\sim N(2250,720),\quad Y\sim N(3860,1050).\]

Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\). [5]

The volumes, in millilitres, of large and small cups of tea are modelled by \(N(200,30)\) and \(N(110,20)\) respectively.

(a) Find the probability that the total volume of a randomly chosen large cup and a randomly chosen small cup is less than 300 ml. [4]

(b) Find the probability that the volume of a randomly chosen large cup is more than twice the volume of a randomly chosen small cup. [6]