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

7 A shop sells old video tapes, of which 1 in 5 on average are known to be damaged.

1. A random sample of 15 tapes is taken. Find the probability that at most 2 are damaged.
2. Find the smallest value of \(n\) if there is a probability of at least 0.85 that a random sample of \(n\) tapes contains at least one damaged tape.
3. A random sample of 1600 tapes is taken. Use a suitable approximation to find the probability that there are at least 290 damaged tapes.

3 A fair dice has four faces. One face is coloured pink, one is coloured orange, one is coloured green and one is coloured black. Five such dice are thrown and the number that fall on a green face are counted. The random variable \(X\) is the number of dice that fall on a green face.

1. Show that the probability of 4 dice landing on a green face is 0.0146 , correct to 4 decimal places.
2. Draw up a table for the probability distribution of \(X\), giving your answers correct to 4 decimal places.

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.

1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.

1. Calculate the value of \(\mu\).
2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.

1 Biscuits are sold in packets of 18. There is a constant probability that any biscuit is broken, independently of other biscuits. The mean number of broken biscuits in a packet has been found to be 2.7 . Find the probability that a packet contains between 2 and 4 (inclusive) broken biscuits.

1 A biased die was thrown 20 times and the number of 5 s was noted. This experiment was repeated many times and the average number of 5 s was found to be 4.8 . Find the probability that in the next 20 throws the number of 5 s will be less than three.

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.

5 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\frac { 1 } { 4 } \mu\). It is given that \(\mathrm { P } ( X > 20 ) = 0.04\).

1. Find \(\mu\).
2. Find \(\mathrm { P } ( 10 < X < 20 )\).
3. 250 independent observations of \(X\) are taken. Find the probability that at least 235 of them are less than 20.

6 The probability that Sue completes a Sudoku puzzle correctly is 0.75 .

1. Sue attempts \(n\) Sudoku puzzles. Find the least value of \(n\) for which the probability that she completes all \(n\) puzzles correctly is less than 0.06 . Sue attempts 14 Sudoku puzzles every month. The number that she completes successfully is denoted by \(X\).
2. Find the value of \(X\) that has the highest probability. You may assume that this value is one of the two values closest to the mean of \(X\).
3. Find the probability that in exactly 3 of the next 5 months Sue completes more than 11 Sudoku puzzles correctly.

4 In a certain mountainous region in winter, the probability of more than 20 cm of snow falling on any particular day is 0.21 .

1. Find the probability that, in any 7-day period in winter, fewer than 5 days have more than 20 cm of snow falling.
2. For 4 randomly chosen 7-day periods in winter, find the probability that exactly 3 of these periods will have at least 1 day with more than 20 cm of snow falling.

3 In Restaurant Bijoux 13\% of customers rated the food as 'poor', 22\% of customers rated the food as 'satisfactory' and \(65 \%\) rated it as 'good'. A random sample of 12 customers who went for a meal at Restaurant Bijoux was taken.

1. Find the probability that more than 2 and fewer than 12 of them rated the food as 'good'. On a separate occasion, a random sample of \(n\) customers who went for a meal at the restaurant was taken.
2. Find the smallest value of \(n\) for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95.

4 The six faces of a fair die are numbered \(1,1,1,2,3,3\). The score for a throw of the die, denoted by the random variable \(W\), is the number on the top face after the die has landed.

1. Find the mean and standard deviation of \(W\).
2. The die is thrown twice and the random variable \(X\) is the sum of the two scores. Draw up a probability distribution table for \(X\).
3. The die is thrown \(n\) times. The random variable \(Y\) is the number of times that the score is 3 . Given that \(\mathrm { E } ( Y ) = 8\), find \(\operatorname { Var } ( Y )\).

5 Suzanne has 20 pairs of shoes, some of which have designer labels. She has 6 pairs of high-heeled shoes, of which 2 pairs have designer labels. She has 4 pairs of low-heeled shoes, of which 1 pair has designer labels. The rest of her shoes are pairs of sports shoes. Suzanne has 8 pairs of shoes with designer labels in total.

1. Copy and complete the table below to show the number of pairs in each category.

   	Designer labels	No designer labels	Total
   High-heeled shoes
   Low-heeled shoes
   Sports shoes
   Total			20

   Suzanne chooses 1 pair of shoes at random to wear.
2. Find the probability that she wears the pair of low-heeled shoes with designer labels.
3. Find the probability that she wears a pair of sports shoes.
4. Find the probability that she wears a pair of high-heeled shoes, given that she wears a pair of shoes with designer labels.
5. State with a reason whether the events 'Suzanne wears a pair of shoes with designer labels' and 'Suzanne wears a pair of sports shoes' are independent. Suzanne chooses 1 pair of shoes at random each day.
6. Find the probability that Suzanne wears a pair of shoes with designer labels on at most 4 days out of the next 7 days.

5 Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The random variable \(X\) is the number of these 12 integers which are multiples of 5 .

1. State the distribution of \(X\) and give its parameters.
2. Calculate the probability that \(X\) is between 3 and 5 inclusive. Fiona now produces \(n\) random integers between 7 and 21 inclusive.
3. Find the least possible value of \(n\) if the probability that none of these integers is a multiple of 5 is less than 0.01.

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.

3

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 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 A box contains 2 green sweets and 5 blue sweets. Two sweets are taken at random from the box, without replacement. The random variable \(X\) is the number of green sweets taken. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 A particular type of bird lays 1,2,3 or 4 eggs in a nest each year. The probability of \(x\) eggs is equal to \(k x\), where \(k\) is a constant.

1. Draw up a probability distribution table, in terms of \(k\), for the number of eggs laid in a year and find the value of \(k\).
2. Find the mean and variance of the number of eggs laid in a year by this type of bird.

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$.

3 Two ordinary fair dice are thrown. The resulting score is found as follows.

• If the two dice show different numbers, the score is the smaller of the two numbers.
• If the two dice show equal numbers, the score is 0 .
  1. Draw up the probability distribution table for the score.
  2. Calculate the expected score.