Combined binomial and normal events

6 In Questa, 60\% of the adults travel to work by car.

1. A random sample of 12 adults from Questa is taken. Find the probability that the number who travel to work by car is less than 10 .
2. A random sample of 150 adults from Questa is taken. Use an approximation to find the probability that the number who travel to work by car is less than 81 .
3. Justify the use of your approximation in part (b).

5 Every day Richard takes a flight between Astan and Bejin. On any day, the probability that the flight arrives early is 0.15 , the probability that it arrives on time is 0.55 and the probability that it arrives late is 0.3 .

1. Find the probability that on each of 3 randomly chosen days, Richard's flight does not arrive late.
2. Find the probability that for 9 randomly chosen days, Richard's flight arrives early at least 3 times.
3. 60 days are chosen at random. Use an approximation to find the probability that Richard's flight arrives early at least 12 times.

5 In Greenton, 70\% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
[0pt]

1. Find the probability that the number of adults in this sample who own a car is less than 6 . [3]
   A random sample of 120 adults from Greenton is now chosen.
2. Use an approximation to find the probability that more than 75 of them own a car.

4 The 1300 train from Jahor to Keman runs every day. The probability that the train arrives late in Keman is 0.35 .

1. For a random sample of 7 days, find the probability that the train arrives late on fewer than 3 days.
   A random sample of 142 days is taken.
2. Use an approximation to find the probability that the train arrives late on more than 40 days.

2 The residents of Persham were surveyed about the reliability of their internet service. 12\% rated the service as 'poor', \(36 \%\) rated it as 'satisfactory' and \(52 \%\) rated it as 'good'. A random sample of 8 residents of Persham is chosen.

1. Find the probability that more than 2 and fewer than 8 of them rate their internet service as poor or satisfactory.
   A random sample of 125 residents of Persham is now chosen.
2. Use an approximation to find the probability that more than 72 of these residents rate their internet service as good.

6 At a company's call centre, \(90 \%\) of callers are connected immediately to a representative.
A random sample of 12 callers is chosen.

1. Find the probability that fewer than 10 of these callers are connected immediately.
   A random sample of 80 callers is chosen.
2. Use an approximation to find the probability that more than 69 of these callers are connected immediately.
3. Justify the use of your approximation in part (b).

4 Kamal has 30 hens. The probability that any hen lays an egg on any day is 0.7 . Hens do not lay more than one egg per day, and the days on which a hen lays an egg are independent.

1. Calculate the probability that, on any particular day, Kamal's hens lay exactly 24 eggs.
2. Use a suitable approximation to calculate the probability that Kamal's hens lay fewer than 20 eggs on any particular day.

7 A survey of adults in a certain large town found that \(76 \%\) of people wore a watch on their left wrist, \(15 \%\) wore a watch on their right wrist and \(9 \%\) did not wear a watch.

1. A random sample of 14 adults was taken. Find the probability that more than 2 adults did not wear a watch.
2. A random sample of 200 adults was taken. Using a suitable approximation, find the probability that more than 155 wore a watch on their left wrist.

3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.

1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.

6

1. In a certain country, \(68 \%\) of households have a printer. Find the probability that, in a random sample of 8 households, 5, 6 or 7 households have a printer.
2. Use an approximation to find the probability that, in a random sample of 500 households, more than 337 households have a printer.
3. Justify your use of the approximation in part (ii).

5 In Pelmerdon 22\% of families own a dishwasher.

1. Find the probability that, of 15 families chosen at random from Pelmerdon, between 4 and 6 inclusive own a dishwasher.
2. A random sample of 145 families from Pelmerdon is chosen. Use a suitable approximation to find the probability that more than 26 families own a dishwasher.

6

1. A manufacturer of biscuits produces 3 times as many cream ones as chocolate ones. Biscuits are chosen randomly and packed into boxes of 10 . Find the probability that a box contains equal numbers of cream biscuits and chocolate biscuits.
2. A random sample of 8 boxes is taken. Find the probability that exactly 1 of them contains equal numbers of cream biscuits and chocolate biscuits.
3. A large box of randomly chosen biscuits contains 120 biscuits. Using a suitable approximation, find the probability that it contains fewer than 35 chocolate biscuits.

7 A manufacturer makes two sizes of elastic bands: large and small. \(40 \%\) of the bands produced are large bands and \(60 \%\) are small bands. Assuming that each pack of these elastic bands contains a random selection, calculate the probability that, in a pack containing 20 bands, there are

1. equal numbers of large and small bands,
2. more than 17 small bands. An office pack contains 150 elastic bands.
3. Using a suitable approximation, calculate the probability that the number of small bands in the office pack is between 88 and 97 inclusive.

6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that

1. 2 pears and 1 orange are eaten, in any order,
2. the third fruit eaten is an orange,
3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
4. Using a suitable approximation, find the probability that fewer than 39 are pears.

7 In Marumbo, three quarters of the adults own a cell phone.

1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive.
2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone.
3. Justify the use of your approximation in part (ii).

7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.

1. Find the probability of throwing a 3 .
2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.

3 On any day at noon, the probabilities that Kersley is asleep or studying are 0.2 and 0.6 respectively.

1. Find the probability that, in any 7-day period, Kersley is either asleep or studying at noon on at least 6 days.
2. Use an approximation to find the probability that, in any period of 100 days, Kersley is asleep at noon on at most 30 days.

5 Blank CDs are packed in boxes of 30 . The probability that a blank CD is faulty is 0.04 . A box is rejected if more than 2 of the blank CDs are faulty.

1. Find the probability that a box is rejected.
2. 280 boxes are chosen randomly. Use an approximation to find the probability that at least 30 of these boxes are rejected.

4 In Quarendon, \(66 \%\) of households are satisfied with the speed of their wifi connection.

1. Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
2. A random sample of 150 households in Quarendon is chosen. Use a suitable approximation to find the probability that more than 84 are satisfied with the speed of their wifi connection. [5]

5

1. Six prizes are allocated, using random numbers, to a group of 12 girls and 8 boys. Calculate the probability that exactly 4 of the prizes are allocated to girls if
   (a) the same child may win more than one prize,
   (b) no child may win more than one prize.
2. Sixty prizes are allocated, using random numbers, to a group of 1200 girls and 800 boys. Use a suitable approximation to calculate the probability that at least 30 of the prizes are allocated to girls. Does it affect your calculation whether or not the same child may win more than one prize? Justify your answer.

4

1. Explain briefly what is meant by a random sample. Random numbers are used to select, with replacement, a sample of size \(n\) from a population numbered 000, 001, 002, ..., 799.
2. If \(n = 6\), find the probability that exactly 4 of the selected sample have numbers less than 500 .
3. If \(n = 60\), use a suitable approximation to calculate the probability that at least 40 of the selected sample have numbers less than 500 .

1. George throws a ball at a target 15 times.

Each time George throws the ball, the probability of the ball hitting the target is 0.48
The random variable \(X\) represents the number of times George hits the target in 15 throws.

1. Find
   1. \(\mathrm { P } ( X = 3 )\)
   2. \(\mathrm { P } ( X \geqslant 5 )\) George now throws the ball at the target 250 times.
2. Use a normal approximation to calculate the probability that he will hit the target more than 110 times.

1. The probability of a leaf cutting successfully taking root is 0.05

Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be

1. 1. exactly 1
   2. more than 2 A second random sample of 160 leaf cuttings is selected.
2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.

1. In a large population \(40 \%\) of adults use online banking.

A random sample of 50 adults is taken.
The random variable \(X\) represents the number of adults in the sample that use online banking.

1. Find
   1. \(\mathrm { P } ( X = 26 )\)
   2. \(\mathrm { P } ( X \geqslant 26 )\)
   3. the smallest value of \(k\) such that \(\mathrm { P } ( X \leqslant k ) > 0.4\) A random sample of 600 adults is taken.
2. 1. Find, using a normal approximation, the probability that no more than 222 of these 600 adults use online banking.
   2. Explain why a normal approximation is suitable in part (b)(i)

4. A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that \(20 \%\) of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3,
2. fewer than 2 . One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,
3. use a suitable approximation to find the probability that there are enough first class stamps.
4. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid.