Independent binomial samples with compound probability

5 In a certain area in the Arctic the probability that it snows on any given day is 0.7 , independent of all other days.

1. Find the probability that in a week (7 days) it snows on at least five days.
   A week in which it snows on at least five days out of seven is called a 'white' week.
2. Find the probability that in three randomly chosen weeks at least one is a white week.
   In a different area in the Arctic, the probability that a week is a white week is 0.8 .
3. Use a suitable approximation to find the probability that in 60 randomly chosen weeks fewer than 47 are white weeks.

2 In a certain country, the probability of more than 10 cm of rain on any particular day is 0.18 , independently of the weather on any other day.

1. Find the probability that in any randomly chosen 7-day period, more than 2 days have more than 10 cm of rain.
2. For 3 randomly chosen 7-day periods, find the probability that exactly two of these periods have at least one day with more than 10 cm of rain.

4 In a certain mountainous region in winter, the probability of more than 20 cm of snow falling on any particular day is 0.21 .

1. Find the probability that, in any 7-day period in winter, fewer than 5 days have more than 20 cm of snow falling.
2. For 4 randomly chosen 7-day periods in winter, find the probability that exactly 3 of these periods will have at least 1 day with more than 20 cm of snow falling.

7 Froox sweets are packed into tubes of 10 sweets, chosen at random. \(25 \%\) of Froox sweets are yellow.

1. Find the probability that in a randomly selected tube of Froox sweets there are
   1. exactly 3 yellow sweets,
   2. at least 3 yellow sweets.
   3. Find the probability that in a box containing 6 tubes of Froox sweets, there is at least 1 tube that contains at least 3 yellow sweets.

5. A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05 . Eggs are packed in boxes of 12 . Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one,
2. more than three. A customer bought three boxes.
3. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. The farmer delivered 10 boxes to a local shop.
4. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
5. Find the probability that a randomly chosen egg weighs more than 68 g .

3. For a particular type of plant \(45 \%\) have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability that this batch contains

1. exactly 5 plants with white flowers,
2. more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batches of plants.
3. Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50 .
4. Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers.

2. The probability of a bolt being faulty is 0.3 . Find the probability that in a random sample of 20 bolts there are

1. exactly 2 faulty bolts,
2. more than 3 faulty bolts. These bolts are sold in bags of 20. John buys 10 bags.
3. Find the probability that exactly 6 of these bags contain more than 3 faulty bolts.

6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is \(5 \%\). If a tray is selected at random, find

1. the probability that at least two of the apples in it are blemished,
2. the probability that exactly two are blemished. Trays are now packed in boxes of 50 trays each. In one such box, find
3. the probability that at most one tray contains at least two blemished apples,
4. the expected number of trays containing at least two blemished apples.
5. Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.

A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05. Eggs are packed in boxes of 12. Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one, [3]
2. more than three. [2]

A customer bought three boxes.

1. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. [3]

The farmer delivered 10 boxes to a local shop.

1. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. [4]

The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g.

1. Find the probability that a randomly chosen egg weighs more than 68 g. [3]

On a production line, bags are filled with cement and weighed as they emerge. It is found that 20\% of the bags are underweight. In a random sample consisting of \(n\) bags, the variance of the number of underweight bags is found to be 2.4.

1. Show that \(n = 15\). [2 marks]
2. Use cumulative binomial probability tables to find the probability that, in a further random sample of 15 bags, the number that are underweight is
   1. less than 3, [3 marks]
   2. at least 5. [2 marks]

Ten samples of 15 bags each are tested. Find the probability that

1. all these batches contain less than 5 underweight bags, [3 marks]
2. the fourth batch tested is the first to contain less than 5 underweight bags. [3 marks]

Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.

1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. [2 marks]
2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions. [2 marks]
3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid. [2 marks]

The probability that Janice sees a kingfisher on any particular day is 0.3. She notes the number, \(X\), of days in a week on which she sees a kingfisher.

1. State one necessary condition for \(X\) to have a binomial distribution. [1]

Assume now that \(X\) has a binomial distribution.

1. Find the probability that, in a week, Janice sees a kingfisher on exactly 2 days. [1]

Each week Janice notes the number of days on which she sees a kingfisher.

1. Find the probability that Janice sees a kingfisher on exactly 2 days in a week during at least 4 of 6 randomly chosen weeks. [3]

A retail bakery makes cherry muffins where, due to the production process, 15% of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.

1. State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid. [4]
2. Using your model from part (a), find the probability that a randomly selected box contains:
   1. exactly 3 muffins with a lower than expected quantity of cherries, [2]
   2. at least 5 muffins with a lower than expected quantity of cherries. [2]
3. The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries. [3]

At a factory that makes crockery the quality control department has found that 10\% of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. [4]

Assume now that your model is valid.

1. Find
   1. P\((X = 3)\), [2]
2. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3. [4]