2.04d Normal approximation to binomial

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.

6 The probability that New Year's Day is on a Saturday in a randomly chosen year is \(\frac { 1 } { 7 }\).

1. 15 years are chosen randomly. Find the probability that at least 3 of these years have New Year's Day on a Saturday.
2. 56 years are chosen randomly. Use a suitable approximation to find the probability that more than 7 of these years have New Year's Day on a Saturday.

7 A die is biased so that the probability of throwing a 5 is 0.75 and the probabilities of throwing a 1,2 , 3 , 4 or 6 are all equal.

1. The die is thrown three times. Find the probability that the result is a 1 followed by a 5 followed by any even number.
2. Find the probability that, out of 10 throws of this die, at least 8 throws result in a 5 .
3. The die is thrown 90 times. Using an appropriate approximation, find the probability that a 5 is thrown more than 60 times.

2 In Scotland, in November, on average \(80 \%\) of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.

1. Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).
2. Give a reason why the use of a normal approximation is justified.

2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.

4 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.

1. Find the probability that at least 2 of the 5 integers are less than or equal to 4 . Robert now generates \(n\) random integers between 1 and 9 inclusive. The random variable \(X\) is the number of these \(n\) integers which are less than or equal to a certain integer \(k\) between 1 and 9 inclusive. It is given that the mean of \(X\) is 96 and the variance of \(X\) is 32 .
2. Find the values of \(n\) and \(k\).

4 In a certain country, on average one student in five has blue eyes.

1. For a random selection of \(n\) students, the probability that none of the students has blue eyes is less than 0.001 . Find the least possible value of \(n\).
2. For a random selection of 120 students, find the probability that fewer than 33 have blue eyes.

7 The time Rafa spends on his homework each day in term-time has a normal distribution with mean 1.9 hours and standard deviation \(\sigma\) hours. On \(80 \%\) of these days he spends more than 1.35 hours on his homework.

1. Find the value of \(\sigma\).
2. Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.
3. A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than 1.35 hours on his homework is between 163 and 173 inclusive.

2 There is a probability of \(\frac { 1 } { 7 }\) that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.

1. Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.
2. Give a reason why the use of a normal approximation is justified.

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).

3 On a production line making cameras, the probability of a randomly chosen camera being substandard is 0.072 . A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard.

5 Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm . The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm .

1. A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
2. 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.
3. Justify the use of your approximation.

2 When visiting the dentist the probability of waiting less than 5 minutes is 0.16 , and the probability of waiting less than 10 minutes is 0.88 .

1. Find the probability of waiting between 5 and 10 minutes. A random sample of 180 people who visit the dentist is chosen.
2. Use a suitable approximation to find the probability that more than 115 of these people wait between 5 and 10 minutes.

2 The probability that George goes swimming on any day is \(\frac { 1 } { 3 }\). Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.

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.

7 In a certain country, \(60 \%\) of mobile phones sold are made by Company \(A , 35 \%\) are made by Company \(B\) and 5\% are made by other companies.

1. Find the probability that, out of a random sample of 13 people who buy a mobile phone, fewer than 11 choose a mobile phone made by Company \(A\).
2. Use a suitable approximation to find the probability that, out of a random sample of 130 people who buy a mobile phone, at least 50 choose a mobile phone made by Company \(B\).
3. A random sample of \(n\) mobile phones sold is chosen. The probability that at least one of these phones is made by Company \(B\) is more than 0.98 . Find the least possible value of \(n\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.

1. Find the probability that an apple selected at random can be used as a toffee apple.
2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.

5 In a certain country the probability that a child owns a bicycle is 0.65 .

1. A random sample of 15 children from this country is chosen. Find the probability that more than 12 own a bicycle.
2. A random sample of 250 children from this country is chosen. Use a suitable approximation to find the probability that fewer than 179 own a bicycle.

5 On average, \(34 \%\) of the people who go to a particular theatre are men.

1. A random sample of 14 people who go to the theatre is chosen. Find the probability that at most 2 people are men.
2. Use an approximation to find the probability that, in a random sample of 600 people who go to the theatre, fewer than 190 are men.

3 It is found that \(10 \%\) of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.

6 The results of a survey by a large supermarket show that \(35 \%\) of its customers shop online.

1. Six customers are chosen at random. Find the probability that more than three of them shop online.
2. For a random sample of \(n\) customers, the probability that at least one of them shops online is greater than 0.95 . Find the least possible value of \(n\).
3. For a random sample of 100 customers, use a suitable approximating distribution to find the probability that more than 39 shop online.

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

1. State two conditions which must be satisfied for a situation to be modelled by a binomial distribution. In a certain village 28\% of all cars are made by Ford.
2. 14 cars are chosen randomly in this village. Find the probability that fewer than 4 of these cars are made by Ford.
3. A random sample of 50 cars in the village is taken. Estimate, using a normal approximation, the probability that more than 18 cars are made by Ford.

7 In tests on a new type of light bulb it was found that the time they lasted followed a normal distribution with standard deviation 40.6 hours. 10\% lasted longer than 5130 hours.

1. Find the mean lifetime, giving your answer to the nearest hour.
2. Find the probability that a light bulb fails to last for 5000 hours.
3. A hospital buys 600 of these light bulbs. Using a suitable approximation, find the probability that fewer than 65 light bulbs will last longer than 5130 hours.

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.