2.04c Calculate binomial probabilities

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1. State three conditions which must be satisfied for a situation to be modelled by a binomial distribution. George wants to invest some of his monthly salary. He invests a certain amount of this every month for 18 months. For each month there is a probability of 0.25 that he will buy shares in a large company, there is a probability of 0.15 that he will buy shares in a small company and there is a probability of 0.6 that he will invest in a savings account.
2. Find the probability that George will buy shares in a small company in at least 3 of these 18 months.

1 In a certain country \(12 \%\) of houses have solar heating. 19 houses are chosen at random. Find the probability that fewer than 4 houses have solar heating.

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.

5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.

1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.

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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 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard deviation 6.5. A book with a height of more than 29 cm is classified as 'large'.

1. Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.
2. \(n\) books are chosen at random. The probability of there being at least 1 large book is more than 0.98 . Find the least possible value of \(n\).

2 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The random variable \(X\) is the score when the die is thrown. The following is the probability distribution table for \(X\). The die is thrown 3 times. Find the probability that the score is 4 on not more than 1 of the 3 throws.

4 When people visit a certain large shop, on average \(34 \%\) of them do not buy anything, \(53 \%\) spend less than \(\\) 50\( and \)13 \%\( spend at least \)\\( 50\).

1. 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of them buy something.
2. \(n\) people visiting the shop are chosen at random. The probability that none of them spends at least \(\\) 50\( is less than 0.04 . Find the smallest possible value of \)n$.

7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.

1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.

7 During the school holidays, each day Khalid either rides on his bicycle with probability 0.6 , or on his skateboard with probability 0.4 . Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is 0.05 . If he rides on his skateboard the probability that he hurts himself is 0.75 .

1. Find the probability that Khalid hurts himself on any particular day.
2. Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard.
3. There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle.
4. Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays.

5 Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week (7 days) is denoted by \(X\).

1. State two conditions that are required for \(X\) to have a binomial distribution.
   On average, Hebe completes 7 out of 10 of these puzzles.
2. Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.
3. Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.

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.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( - 3 , \sigma ^ { 2 } \right)\). The probability that a randomly chosen value of \(X\) is positive is 0.25 .

1. Find the value of \(\sigma\).
2. Find the probability that, of 8 random values of \(X\), fewer than 2 will be positive.

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.

3 The probability that Janice will buy an item online in any week is 0.35 . Janice does not buy more than one item online in any week.

1. Find the probability that, in a 10 -week period, Janice buys at most 7 items online.
2. The probability that Janice buys at least one item online in a period of \(n\) weeks is greater than 0.99 . Find the smallest possible value of \(n\).

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.

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1. The lengths, in centimetres, of middle fingers of women in Raneland have a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(25 \%\) of these women have fingers longer than 8.8 cm and \(17.5 \%\) have fingers shorter than 7.7 cm .
   1. Find the values of \(\mu\) and \(\sigma\).
      The lengths, in centimetres, of middle fingers of women in Snoland have a normal distribution with mean 7.9 and standard deviation 0.44. A random sample of 5 women from Snoland is chosen.
   2. Find the probability that exactly 3 of these women have middle fingers shorter than 8.2 cm .
2. The random variable \(X\) has a normal distribution with mean equal to the standard deviation. Find the probability that a particular value of \(X\) is less than 1.5 times the mean.

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.

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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.

4 Single cards, chosen at random, are given away with bars of chocolate. Each card shows a picture of one of 20 different football players. Richard needs just one picture to complete his collection. He buys 5 bars of chocolate and looks at all the pictures. Find the probability that

1. Richard does not complete his collection,
2. he has the required picture exactly once,
3. he completes his collection with the third picture he looks at.

7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.

1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
   \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).

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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.

5 A box contains 300 discs of different colours. There are 100 pink discs, 100 blue discs and 100 orange discs. The discs of each colour are numbered from 0 to 99 . Five discs are selected at random, one at a time, with replacement. Find

1. the probability that no orange discs are selected,
2. the probability that exactly 2 discs with numbers ending in a 6 are selected,
3. the probability that exactly 2 orange discs with numbers ending in a 6 are selected,
4. the mean and variance of the number of pink discs selected.

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.