Multiple probability calculations

6. The random variable \(X\) has a normal distribution with mean 20 and standard deviation 4 .

1. Find \(\mathrm { P } ( X > 25 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( 20 < X < d ) = 0.4641\)

7

1. The weight, \(X\) grams, of soup in a carton may be modelled by a normal random variable with mean 406 and standard deviation 4.2. Find the probability that the weight of soup in a carton:
   1. is less than 400 grams;
   2. is between 402.5 grams and 407.5 grams.
2. The weight, \(Y\) grams, of chopped tomatoes in a tin is a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).
   1. Given that \(\mathrm { P } ( Y < 310 ) = 0.975\), explain why: $$310 - \mu = 1.96 \sigma$$
   2. Given that \(\mathrm { P } ( Y < 307.5 ) = 0.86\), find, to two decimal places, values for \(\mu\) and \(\sigma\).
      (4 marks)

1 In large-scale tree-felling operations, a machine cuts down trees, strips off the branches and then cuts the trunks into logs of length \(X\) metres for transporting to a sawmill. It may be assumed that values of \(X\) are normally distributed with mean \(\mu\) and standard deviation 0.16 , where \(\mu\) can be set to a specific value.

1. Given that \(\mu\) is set to 3.3 , determine:
   1. \(\mathrm { P } ( X < 3.5 )\);
   2. \(\mathrm { P } ( X > 3.0 )\);
   3. \(\mathrm { P } ( 3.0 < X < 3.5 )\).
2. The sawmill now requires a batch of logs such that there is a probability of 0.025 that any given log will have a length less than 3.1 metres. Determine, to two decimal places, the new value of \(\mu\).

3 UPVC facia board is supplied in lengths labelled as 5 metres. The actual length, \(X\) metres, of a board may be modelled by a normal distribution with a mean of 5.08 and a standard deviation of 0.05 .

1. Determine:
   1. \(\mathrm { P } ( X < 5 )\);
   2. \(\mathrm { P } ( 5 < X < 5.10 )\).
2. Determine the probability that the mean length of a random sample of 4 boards:
   1. exceeds 5.05 metres;
   2. is exactly 5 metres.
3. Assuming that the value of the standard deviation remains unchanged, determine the mean length necessary to ensure that only 1 per cent of boards have lengths less than 5 metres.

2 The volume of Everwhite toothpaste in a pump-action dispenser may be modelled by a normal distribution with a mean of 106 ml and a standard deviation of 2.5 ml . Determine the probability that the volume of Everwhite in a randomly selected dispenser is:

1. less than 110 ml ;
2. more than 100 ml ;
3. between 104 ml and 108 ml ;
4. not exactly 106 ml .

2 The weight, \(X\) grams, of the contents of a tin of baked beans can be modelled by a normal random variable with a mean of 421 and a standard deviation of 2.5.

1. Find:
   1. \(\mathrm { P } ( X = 421 )\);
   2. \(\mathrm { P } ( X < 425 )\);
   3. \(\mathrm { P } ( 418 < X < 424 )\).
2. Determine the value of \(x\) such that \(\mathrm { P } ( X < x ) = 0.98\).
3. The weight, \(Y\) grams, of the contents of a tin of ravioli can be modelled by a normal random variable with a mean of \(\mu\) and a standard deviation of 3.0 . Find the value of \(\mu\) such that \(\mathrm { P } ( Y < 410 ) = 0.01\).

3. The random variable \(X\) is normally distributed with a mean of 42 and a variance of 18 . Find

1. \(\mathrm { P } ( X \leq 45 )\),
2. \(\mathrm { P } ( 32 \leq X \leq 38 )\),
3. the value of \(x\) such that \(\mathrm { P } ( X \leq x ) = 0.95\)

4. The random variable \(A\) is normally distributed with a mean of 32.5 and a variance of 18.6 Find

1. \(\mathrm { P } ( A < 38.2 )\),
2. \(\mathrm { P } ( 31 \leq A \leq 35 )\), The random variable \(B\) is normally distributed with a standard deviation of 7.2
   Given also that \(\mathrm { P } ( B > 110 ) = 0.138\),
3. find the mean of \(B\).

2.

1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
   (a) Find \(\mathrm { P } ( X > 25 )\).
   (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
   (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).

4.

1. The masses, in grams, of plums of a certain kind have the distribution \(\mathrm { N } ( 55,18 )\). The heaviest \(5 \%\) of plums are classified as extra large. Find the minimum mass of extra large plums.
2. The masses, in grams, of apples of a certain kind have the distribution \(\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)\). It is given that half of the apples have masses between 62 g and 72 g . Determine \(\sigma\).

8 The masses, \(X\) grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and \(70 \%\) of the tomatoes have masses greater than 53.0 g .

1. Find the percentage of tomatoes with masses greater than 59.0 g .
2. Find the percentage of tomatoes with masses greater than 65.0 g .
3. Given that \(\mathrm { P } ( a < X < 50 ) = 0.1\), find \(a\).

6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.

1. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
2. \(20 \%\) of employees take longer than \(t\) minutes to complete the task. Find the value of \(t\).
3. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.

4 Chopped lettuce is sold in bags nominally containing 100 grams.
The weight, \(X\) grams, of chopped lettuce, delivered by the machine filling the bags, may be assumed to be normally distributed with mean \(\mu\) and standard deviation 4.

1. Assuming that \(\mu = 106\), determine the probability that a randomly selected bag of chopped lettuce:
   1. weighs less than 110 grams;
   2. is underweight.
2. Determine the minimum value of \(\mu\) so that at most 2 per cent of bags of chopped lettuce are underweight. Give your answer to one decimal place.
3. Boxes each contain 10 bags of chopped lettuce. The mean weight of a bag of chopped lettuce in a box is denoted by \(\bar { X }\). Given that \(\mu = 108.5\) :
   1. write down values for the mean and variance of \(\bar { X }\);
   2. determine the probability that \(\bar { X }\) exceeds 110 .

2 The heights of sunflowers may be assumed to be normally distributed with a mean of 185 cm and a standard deviation of 10 cm .

1. Determine the probability that the height of a randomly selected sunflower:
   1. is less than 200 cm ;
   2. is more than 175 cm ;
   3. is between 175 cm and 200 cm .
2. Determine the probability that the mean height of a random sample of 4 sunflowers is more than 190 cm .

17 The lifetime of Zaple smartphone batteries, \(X\) hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours. 17

1. 1. Find \(\mathrm { P } ( X \neq 8 )\) 17
2. (ii) Find \(\mathrm { P } ( 6 < X < 10 )\)
   17
3. Determine the lifetime exceeded by \(90 \%\) of Zaple smartphone batteries.
   17
4. A different smartphone, Kaphone, has its battery's lifetime, \(Y\) hours, modelled by a normal distribution with mean 7 hours and standard deviation \(\sigma\). 25\% of randomly selected Kaphone batteries last less than 5 hours.
   Find the value of \(\sigma\), correct to three significant figures.

16 A farm supplies apples to a supermarket.
The diameters of the apples, \(D\) centimetres, are normally distributed with mean 6.5 and standard deviation 0.73
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1. 1. 
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2. (ii) Find \(\mathrm { P } ( D > 7 )\)
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   16 (iii) The supermarket only accepts apples with diameters between 5 cm and 8 cm . Find the proportion of apples that the supermarket accepts.

   [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   
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3. The farm also supplies plums to the supermarket. These plums have diameters that are normally distributed.
   It is found that \(60 \%\) of these plums have a diameter less than 5.9 cm .
   It is found that \(20 \%\) of these plums have a diameter greater than 6.1 cm .
   Find the mean and standard deviation of the diameter, in centimetres, of the plums supplied by the farm.