Mixed sum threshold probability

3 Mary buys 3 packets of sugar and 5 packets of coffee and puts them in her shopping basket, together with her purse which weighs 350 g . Weights of packets of sugar are normally distributed with mean 500 g and standard deviation 20 g . Weights of packets of coffee are normally distributed with mean 200 g and standard deviation 12 g . Find the probability that the total weight in the shopping basket is less than 2900 g .

3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.

4 The masses, in grams, of tomatoes of type \(A\) and type \(B\) have the distributions \(\mathrm { N } \left( 125,30 ^ { 2 } \right)\) and \(\mathrm { N } \left( 130,32 ^ { 2 } \right)\) respectively.

1. Find the probability that the total mass of 4 randomly chosen tomatoes of type \(A\) and 6 randomly chosen tomatoes of type \(B\) is less than 1.5 kg .
2. Find the probability that a randomly chosen tomato of type \(A\) has a mass that is at least \(90 \%\) of the mass of a randomly chosen tomato of type \(B\).

3 In the manufacture of child car seats, a resin made up of three ingredients is used. The ingredients are two polymers and an impact modifier. The resin is prepared in batches. Each ingredient is supplied by a separate feeder and the amount supplied to each batch, in kg, is assumed to be Normally distributed with mean and standard deviation as shown in the table below. The three feeders are also assumed to operate independently of each other.

1. Find the probability that, in a randomly chosen batch of resin, there is no more than 2100 kg of polymer 1.
2. Find the probability that, in a randomly chosen batch of resin, the amount of polymer 1 exceeds the amount of polymer 2 by at least 400 kg .
3. Find the value of \(b\) such that the total amount of the ingredients in a randomly chosen batch exceeds \(b \mathrm {~kg} 95 \%\) of the time.
4. Polymer 1 costs \(\pounds 1.20\) per kg, polymer 2 costs \(\pounds 1.30\) per kg and the impact modifier costs \(\pounds 0.80\) per kg. Find the mean and variance of the total cost of a batch of resin.
5. Each batch of resin is used to make a large number of car seats from which a random sample of 50 seats is selected in order that the tensile strength (in suitable units) of the resin can be measured. From one such sample, the \(99 \%\) confidence interval for the true mean tensile strength of the resin in that batch was calculated as \(( 123.72,127.38 )\). Find the mean and standard deviation of the sample.

2 The mass \(J \mathrm {~kg}\) of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass Kkg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04 .

1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg .
2. Find the probability that the mass of one bag of King Edward potatoes is more than \(75 \%\) of the mass of one bag of Jersey potatoes.

6 A particular lift has a maximum load capacity of 700 kg .
The weights of men are normally distributed with mean 80 kg and standard deviation 10 kg . The weights of women are normally distributed with mean 69 kg and standard deviation 5 kg . You may assume that weights of people are independent.

1. Find the probability that when 6 men and 3 women are in the lift, the load exceeds 700 kg . A sign in the lift states: "Maximum number of people in the lift is \(c\) "
2. Find the value of \(c\) such that the probability of the load exceeding 700 kg is less than \(2.5 \%\) no matter the gender of the occupants.

7. A company makes cricket balls and tennis balls. The weights of cricket balls, \(C\) grams, follow a normal distribution $$C \sim \mathrm {~N} \left( 160,1.25 ^ { 2 } \right)$$ Three cricket balls are selected at random.

1. Find the probability that their total weight is more than 475.8 grams. The weights of tennis balls, \(T\) grams, follow a normal distribution $$T \sim \mathrm {~N} \left( 60,2 ^ { 2 } \right)$$ Five tennis balls and two cricket balls are selected at random.
2. Find the probability that the total weight of the five tennis balls and the two cricket balls is more than 625 grams. A random sample of \(n\) tennis balls \(T _ { 1 } , T _ { 2 } , \ldots , T _ { n }\) is taken.
   The random variable \(Y = ( n - 1 ) T _ { 1 } - \sum _ { r = 2 } ^ { n } T _ { r }\) Given that \(\mathrm { P } ( Y > 40 ) = 0.0838\) correct to 4 decimal places,
3. find \(n\).
   

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   Q7

3 The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g .

1. Find the probability that the total weight of 5 randomly selected bananas is at least 1 kg . When a banana is peeled the change in its weight is modelled as being a reduction of \(35 \%\).
2. Find the probability that the weight of a randomly selected peeled banana is at most 150 g Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
3. Find the probability that the total weight of a smoothie is less than 700 g .

2 A manufacturer is testing how long coloured LED lights will last before the battery runs out, using two different battery types. The times in hours before the battery runs out are modelled by independent Normal distributions with means and standard deviations as shown in the table.

1. In a particular test, a battery of type A is used and the time taken for it to run out is recorded. This process is repeated until a total of 5 randomly selected batteries have been used. Determine the probability that the total time the 5 batteries last is at least 120 hours.
2. In a similar test, 3 randomly selected batteries of type A are used, one after the other. Then 2 randomly selected batteries of type B are used, one after the other. Determine the probability that the 3 type A batteries last longer in total than the 2 type B batteries.
3. Explain why it is necessary that the Normal distributions are independent in order to be able to find the probability in part (b).

3 A supermarket sells cashew nuts in three different sizes of bag: small, medium and large. The weights in grams of the nuts in each type of bag are modelled by independent Normal distributions as shown in Table 3. \begin{table}[h] \end{table}

1. Find the probability that the mean weight of two randomly selected large bags is at least 200 g .
2. Find the probability that the total weight of eight randomly selected small bags is greater than the total weight of two randomly selected medium bags and one randomly selected large bag.

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.

1. Find the probability that the total mass of these 6 bags of flour is less than 4130 g. [5]
2. Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags. [5]

Ranjit goes to mathematics lectures and physics lectures. The length, in minutes, of a mathematics lecture is modelled by the variable \(X\) with distribution N(36, 3.5²). The length, in minutes, of a physics lecture is modelled by the independent variable \(Y\) with distribution N(55, 5.2²).

1. Find the probability that the total length of two mathematics lectures and one physics lecture is less than 140 minutes. [4]
2. Ranjit calculates how long he will need to spend revising the content of each lecture as follows. Each minute of a mathematics lecture requires 1 minute of revision and each minute of a physics lecture requires 1½ minutes of revision. Find the probability that the total revision time required for one mathematics lecture and one physics lecture is more than 100 minutes. [4]

The mass of waste in filled large dustbin bags is normally distributed with a mean of 6.8 kg and a standard deviation of 1.5 kg. The mass of waste in filled small dustbin bags is normally distributed with a mean of 3.2 kg and a standard deviation of 0.6 kg. One week there are 8 large and 3 small dustbin bags left for collection outside a block of flats. Find the probability that this waste has a total mass of more than 70 kg. [7 marks]