2.04d Normal approximation to binomial

7 On any given day, the probability that Moena messages her friend Pasha is 0.72 .

1. Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days.
2. Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January.
3. Use an approximation to find the probability that in any period of 100 days Moena messages Pasha on fewer than 64 days.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.

1. Find the expected value of \(X\).
2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
3. Find the probability that fewer than 6 throws are required to obtain a pair of tails.
   On a different occasion, a pair of fair coins is thrown 80 times.
4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.

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.

7 In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800 . Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table.

1. 1. Find the probability that a randomly chosen household is in Shan and has poor broadband service.
   2. Find the probability that a randomly chosen household has good broadband service given that the household is in Shan.
      In the whole of Arka there are a large number of households. A survey showed that \(35 \%\) of households in Arka have no broadband service.
2. 1. 10 households in Arka are chosen at random. Find the probability that fewer than 3 of these households have no broadband service.
   2. 120 households in Arka are chosen at random. Use an approximation to find the probability that more than 32 of these households have no broadband service.
      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 Eli has four fair 4 -sided dice with sides labelled \(1,2,3,4\). He throws all four dice at the same time. The random variable \(X\) denotes the number of 2s obtained.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 64 }\).
2. Complete the following probability distribution table for \(X\).

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	\(\frac { 81 } { 256 }\)			\(\frac { 3 } { 64 }\)	\(\frac { 1 } { 256 }\)

3. Find \(\mathrm { E } ( X )\).
   Eli throws the four dice at the same time on 96 occasions.
4. Use an approximation to find the probability that he obtains at least two 2 s on fewer than 20 of these occasions.

5 The lengths of Western bluebirds are normally distributed with mean 16.5 cm and standard deviation 0.6 cm . A random sample of 150 of these birds is selected.

1. How many of these 150 birds would you expect to have length between 15.4 cm and 16.8 cm ?
   The lengths of Eastern bluebirds are normally distributed with mean 18.4 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that \(72 \%\) of Eastern bluebirds have length greater than 17.1 cm .
2. Find the value of \(\sigma\).
   A random sample of 120 Eastern bluebirds is chosen.
3. Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1 cm .

2 Anil is a candidate in an election. He received \(40 \%\) of the votes. A random sample of 120 voters is chosen. Use an approximation to find the probability that, of the 120 voters, between 36 and 54 inclusive voted for Anil.

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.

6 The residents of Mahjing were asked to classify their local bus service:

• \(25 \%\) of residents classified their service as good.
• \(60 \%\) of residents classified their service as satisfactory.
• \(15 \%\) of residents classified their service as poor.
  1. A random sample of 110 residents of Mahjing is chosen.

Use a suitable approximation to find the probability that fewer than 22 residents classified their bus service as good.

For a random sample of 10 residents of Mahjing, find the probability that fewer than 8 classified their bus service as good or satisfactory.

Three residents of Mahjing are selected at random. Find the probability that one resident classified the bus service as good, one as satisfactory and one as poor.

5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
[0pt]

1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.

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.

7 There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table. A student is chosen at random.

1. 1. Find the probability that the student prefers swimming.
   2. Determine whether the events 'the student is male' and 'the student prefers swimming' are independent, justifying your answer.
      On average at all the schools in this country \(30 \%\) of the students do not like any sports.
2. 1. 10 of the students from this country are chosen at random. Find the probability that at least 3 of these students do not like any sports.
   2. 90 students from this country are now chosen at random. Use an approximation to find the probability that fewer than 32 of them do not like any sports.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A company sells small and large bags of rice. The masses of the small bags of rice are normally distributed with mean 1.20 kg and standard deviation 0.16 kg .

1. In a random sample of 500 of these small bags of rice, how many would you expect to have a mass greater than 1.26 kg ?
   The masses of the large bags of rice are normally distributed with mean 2.50 kg and standard deviation \(\sigma \mathrm { kg } .20 \%\) of these large bags of rice have a mass less than 2.40 kg .
2. Find the value of \(\sigma\).
   A random sample of 80 large bags of rice is chosen.
3. Use a suitable approximation to find the probability that fewer than 22 of these large bags of rice have a mass less than 2.40 kg .

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).

2 In a large college, \(32 \%\) of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.

3 A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight. A small egg weighs less than 76 grams and \(40 \%\) of the eggs are classified as medium.

1. Find the percentage of eggs that are classified as small.
2. Find the least possible weight of an egg classified as large.
   150 of the eggs for sale last week were weighed.
3. Use an approximation to find the probability that more than 68 of these eggs were classified as medium.

3 A factory produces a certain type of electrical component. It is known that \(15 \%\) of the components produced are faulty. A random sample of 200 components is chosen. Use an approximation to find the probability that more than 40 of these components are faulty.

5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.

1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
   2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
2. Find the probability that at least three of them will pass tomorrow.
   75 people will take their advanced driving test next week.
   [0pt]
3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]

5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.

1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-09_2717_29_105_22}
2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.

7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.

1. Find \(\mathrm { P } ( X = 5 )\) .
   ............................................................................................................................................
2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
3. Find the probability that Leno scores his second goal on or before his 5th attempt. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017} \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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.

1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight.