All components random including container

4 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 \(S \sim \mathrm {~N} ( 300,45 )\), \(R \sim \mathrm {~N} ( 200,25 )\) and \(\mathrm { B } \sim \mathrm { N } ( 15,4 )\). A full box of Seeds \& Raisins is chosen at random.
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1. Find the probability that the total weight of the box and its contents is more than 500 grams. [5]
2. 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 Packets of cereal are packed in boxes, each containing 6 packets. The masses of the packets are normally distributed with mean 510 g and standard deviation 12 g . The masses of the empty boxes are normally distributed with mean 70 g and standard deviation 4 g .

1. Find the probability that the total mass of a full box containing 6 packets is between 3050 g and 3150 g .
2. A packet and an empty box are chosen at random. Find the probability that the mass of the packet is at least 8 times the mass of the empty box.

7 Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg . The masses of the bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg .

1. Find the probability that the total mass of a full box of 20 bags is less than 20.6 kg .
2. Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg .

1 A laminate consists of 4 layers of material \(C\) and 3 layers of material \(D\). The thickness of a layer of material \(C\) has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material \(D\) has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.

1. Find the mean and variance of the total thickness of the laminate.
2. What total thickness is exceeded by \(1 \%\) of the laminates?

1. A farm produces potatoes. The potatoes are packed into sacks.

The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg
   (6) Sacks of potatoes are randomly selected and packed onto pallets.
   The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg

4. A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg
   
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1. The weights of eggs, \(E\) grams, follow a normal distribution, \(\mathrm { N } \left( 60,3 ^ { 2 } \right)\)

The weights of empty small boxes, \(S\) grams, follow a normal distribution, \(\mathrm { N } \left( 24,1.8 ^ { 2 } \right)\)
The weights of empty large boxes, \(L\) grams, follow a normal distribution, \(\mathrm { N } \left( 40,2.1 ^ { 2 } \right)\)
Small boxes of eggs contain 6 randomly selected eggs.
Large boxes of eggs contain 12 randomly selected eggs.

1. Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.
2. Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.