2.04c Calculate binomial probabilities

2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.

1. Find the probability that it takes fewer than 7 spins for George to obtain a 5 .
   George spins the spinner 10 times.
2. Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.

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]

6 The heights of the female students at Breven college are normally distributed:

• \(90 \%\) of the female students have heights less than 182.7 cm .
• \(40 \%\) of the female students have heights less than 162.5 cm .
  1. Find the mean and the standard deviation of the heights of the female students at Breven college. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-10_2715_41_110_2008} \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-11_2723_35_101_20}

Ten female students are chosen at random from those at Breven college.

Find the probability that fewer than 8 of these 10 students have heights more than 162.5 cm .

1 At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports. One student is chosen at random from the 120 students. Events \(X\) and \(N\) are defined as follows: \(X\) : the student is in Year 1 \(N\) : the student plays netball.

1. Find \(\mathrm { P } ( X \mid N )\).
2. Find \(\mathrm { P } ( N \mid X )\).
3. Determine whether or not \(X\) and \(N\) are independent events.
   One of the students who plays netball takes 8 shots at goal. On each shot, the probability that she will succeed is 0.15 , independently of all other shots.
4. Find the probability that she succeeds on fewer than 3 of these shots.

5 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.

1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
   Jake chooses chocolates one at a time at random from the production line.
2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.

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.

3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.

1. Draw up the probability distribution table for the number of jellies that Jemeel chooses.
   The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
2. Find the probability that no more than 7 of these boxes contain more jellies than chocolates.

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.