2.04b Binomial distribution: as model B(n,p)

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.

7

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.

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.

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

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.

6 On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65 , independently of all other occasions.

1. Find the probability that she will perform the routine correctly on exactly 5 occasions out 7 .
2. On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.
3. On another day she performs the routine \(n\) times. Find the smallest value of \(n\) for which the expected number of correct performances is at least 8 .

7 A fair die has one face numbered 1, one face numbered 3, two faces numbered 5 and two faces numbered 6 .

1. Find the probability of obtaining at least 7 odd numbers in 8 throws of the die. The die is thrown twice. Let \(X\) be the sum of the two scores. The following table shows the possible values of \(X\). \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Second throw}

   		1	3	5	5	6	6
   \cline { 2 - 8 }	1	2	4	6	6	7	7
   	3	4	6	8	8	9	9
   First	5	6	8	10	10	11	11
   throw	5	6	8	10	10	11	11
   	6	7	9	11	11	12	12
   	6	7	9	11	11	12	12

   \end{table}
2. Draw up a table showing the probability distribution of \(X\).
3. Calculate \(\mathrm { E } ( X )\).
4. Find the probability that \(X\) is greater than \(\mathrm { E } ( X )\).

1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.

7 The weights, \(X\) grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.

1. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 128 ) = 0.7465\).
3. Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.

3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.

1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.

6

1. State three conditions that must be satisfied for a situation to be modelled by a binomial distribution. On any day, there is a probability of 0.3 that Julie's train is late.
2. Nine days are chosen at random. Find the probability that Julie's train is late on more than 7 days or fewer than 2 days.
3. 90 days are chosen at random. Find the probability that Julie's train is late on more than 35 days or fewer than 27 days.

2 The random variable \(X\) is the daily profit, in thousands of dollars, made by a company. \(X\) is normally distributed with mean 6.4 and standard deviation 5.2.

1. Find the probability that, on a randomly chosen day, the company makes a profit between \(\\) 10000\( and \)\\( 12000\).
2. Find the probability that the company makes a loss on exactly 1 of the next 4 consecutive days.

4 The mean of a certain normally distributed variable is four times the standard deviation. The probability that a randomly chosen value is greater than 5 is 0.15 .

1. Find the mean and standard deviation.
2. 200 values of the variable are chosen at random. Find the probability that at least 160 of these values are less than 5 .

5 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 82,126 )\).

1. A value of \(X\) is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .
2. Five independent observations of \(X\) are taken. Find the probability that at most one of them is greater than 87.
3. Find the value of \(k\) such that \(\mathrm { P } ( 87 < X < k ) = 0.3\).