Comparison involving sums or multiples

2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions \(W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )\) and \(Y \sim \mathrm {~N} ( 3860,1050 )\). Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\).

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

3 The lengths of red pencils are normally distributed with mean 6.5 cm and standard deviation 0.23 cm .

1. Two red pencils are chosen at random. Find the probability that their total length is greater than 12.5 cm . The lengths of black pencils are normally distributed with mean 11.3 cm and standard deviation 0.46 cm .
2. Find the probability that the total length of 3 red pencils is more than 6.7 cm greater than the length of 1 black pencil.

6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.

1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.

6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.

1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man. [4]

4 The weights of men follow a normal distribution with mean 71 kg and standard deviation 7 kg . The weights of women follow a normal distribution with mean 57 kg and standard deviation 5 kg . The total weight of 5 men and 2 women chosen randomly is denoted by \(X \mathrm {~kg}\).

1. Show that \(\mathrm { E } ( X ) = 469\) and \(\operatorname { Var } ( X ) = 295\).
2. The total weight of 4 men and 3 women chosen randomly is denoted by \(Y \mathrm {~kg}\). Find the mean and standard deviation of \(X - Y\) and hence find \(\mathrm { P } ( X - Y > 22 )\).

5 Climbing ropes produced by a manufacturer have breaking strengths which are normally distributed with mean 160 kg and standard deviation 11.3 kg . A group of climbers have weights which are normally distributed with mean 66.3 kg and standard deviation 7.1 kg .

1. Find the probability that a rope chosen randomly will break under the combined weight of 2 climbers chosen randomly. Each climber carries, in a rucksack, equipment amounting to half his own weight.
2. Find the mean and variance of the combined weight of a climber and his rucksack.
3. Find the probability that the combined weight of a climber and his rucksack is greater than 87 kg .

6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.

1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man.

4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.

1. Small containers and large containers are independently filled with fruit juice.

The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable \(W\) represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container.
\(W \sim \mathrm {~N} ( a , b )\) where \(a\) and \(b\) are positive constants.

1. Find the value of \(a\) and the value of \(b\)
2. Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
3. Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.

7. The heights, in cm, of the male employees in a large company follow a normal distribution with mean 177 and standard deviation 5 The heights, in cm, of the female employees follow a normal distribution with mean 163 and standard deviation 4 A male employee and a female employee are chosen at random.

1. Find the probability that the male employee is taller than the female employee. Six male employees and four female employees are chosen at random.
2. Find the probability that their total height is less than 17 m .

1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .

One large and 3 small bottles of Blumen are chosen at random.

1. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.

7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.

1. Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
2. Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
3. Find the value of \(d\) so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than \(d\) grams.

6 The length \(L\) of a particular type of fence panel is Normally distributed with mean 179.2 cm and standard deviation 0.8 cm . You should assume that the lengths of individual fence panels are independent of each other.

1. Find the probability that the length of a randomly chosen fence panel is at least 180 cm .
2. Find the probability that the total length of 5 randomly chosen fence panels is less than 895 cm . The width \(W\) of a fence post is Normally distributed with mean 9.8 cm and standard deviation 0.3 cm . A straight fence is constructed using 6 posts and 5 panels with no gaps between them. Fig. 6 shows a view from above of the first two posts, the first panel and the start of the second panel. You should assume that the lengths of fence panels and widths of fence posts are independent. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4caa7409-cb32-41da-ad64-012a45753296-6_213_1522_934_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
3. Determine the probability that the total length of the fence, including the posts, is less than 9.5 m .
4. State another assumption that is necessary for the calculation of the probability in part (c) to be valid.