Binomial Distribution

The random variable \(X\) has the binomial distribution B(16, 0·3). Showing your calculation, find \(P(X = 7)\). [2]

7 James throws a discus repeatedly in an attempt to achieve a successful throw. A throw is counted as successful if the distance achieved is over 40 metres. For each throw, the probability that James is successful is \(\frac { 1 } { 4 }\), independently of all other throws. Find the probability that James takes

1. exactly 5 throws to achieve the first successful throw,
2. more than 8 throws to achieve the first successful throw. In order to qualify for a competition, a discus-thrower must throw over 40 metres within at most six attempts. When a successful throw is achieved, no further throws are taken. Find the probability that James qualifies for the competition. Colin is another discus-thrower. For each throw, the probability that he will achieve a throw over 40 metres is \(\frac { 1 } { 3 }\), independently of all other throws. Find the probability that exactly one of James and Colin qualifies for the competition.

5

1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
   Find the probability that the team contains no more than 3 girls.
2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).

A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3 and 5 inclusive. [3]

A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents' evening. The number of students in the sample who are from years 12 and 13 is denoted by \(X\).

1. State a suitable binomial model for \(X\). [1]

Use your model to answer the following.

1. 1. Write down an expression for \(\text{P}(X = x)\). [1]
   2. State, in set notation, the values of \(x\) for which your expression is valid. [1]
2. Find \(\text{P}(5 \leqslant X \leqslant 9)\). [2]
3. State one disadvantage of using a random sample in this context. [1]

6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter \(p\) equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.

1. Determine the probability that a packet contains:
   1. less than or equal to 15 red pegs;
   2. exactly 10 red pegs;
   3. more than 5 but fewer than 15 red pegs.
2. Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet \(= 10.5\) Variance of number of red pegs per packet \(= 20.41\) Comment on the validity of Sly's claim.

The random variable \(X\) is such that \(X \sim B(n, p)\) The mean value of \(X\) is 225 The variance of \(X\) is 144 Find \(p\). Circle your answer. [1 mark] 0.36 \quad 0.6 \quad 0.64 \quad 0.8

The random variable \(X\) is such that \(X \sim B\left(n, \frac{1}{3}\right)\) The standard deviation of \(X\) is 4 Find the value of \(n\). Circle your answer. [1 mark] 9 \quad 12 \quad 18 \quad 72

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 \(k\).

2 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

2 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

4 At a call centre, \(85 \%\) of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

1. Find the probability that exactly 29 of these callers are put on hold.
2. Find the probability that at least 29 of these callers are put on hold.
3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.

2. The probability of a bolt being faulty is 0.3 . Find the probability that in a random sample of 20 bolts there are

1. exactly 2 faulty bolts,
2. more than 3 faulty bolts. These bolts are sold in bags of 20. John buys 10 bags.
3. Find the probability that exactly 6 of these bags contain more than 3 faulty bolts.

Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.

1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. [2 marks]
2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions. [2 marks]
3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid. [2 marks]

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.

20% of packets of a certain kind of cereal contain a free gift. Jane buys one packet a week for 8 weeks. The number of free gifts that Jane receives is denoted by \(X\). Assuming that Jane's 8 packets can be regarded as a random sample, find

1. P(\(X = 3\)), [3]
2. P(\(X \geqslant 3\)), [2]
3. E(\(X\)). [2]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of \(X\). [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(\(X\)) and Var (\(X\)). [3]

1. A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let \(X\) be the number of correct answers that Sam achieves.

1. State the distribution of \(X\) Let \(M\) be the number of marks that Sam achieves.
2. 1. State the distribution of \(M\) in terms of \(X\)
   2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
3. Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
   Given that the probability that more than \(n\) but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
4. find the value of \(n\)

7. Emma throws a fair coin 15 times and records the number of times it shows a head.
a State the appropriate distribution to model the number of times the coin shows a head giving any relevant parameter values.
b Find the probability that Emma records:
i exactly 8 heads
ii at least 4 heads.

1. A factory produces components. Each component has a unique identity number and it is assumed that \(2 \%\) of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.
   1. Identify a sampling frame.
   The statistic \(F\) represents the number of faulty components in the random sample of size 50.
2. Specify the sampling distribution of \(F\).

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 A survey into left-handedness found that 13\% of the population of the world are left-handed.

1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution \(\mathrm { B } ( 20,0.13 )\).
2. Assuming that this binomial model is appropriate, calculate the probability that fewer than \(13 \%\) of the 20 children are left-handed.

At a fairground, Kirsty throws \(n\) balls in order to try to knock coconuts off their stands. Any coconuts she knocks off are replaced before she throws again. Kirsty counts the number of coconuts she successfully knocks off their stands. On average, she knocks off a coconut with 20\% of her throws.

1. What assumptions are needed in order to model this situation with a binomial distribution? Explain whether these assumptions are reasonable. [2]

Kirsty uses a spreadsheet to produce the following diagrams, showing the probability distributions of the number of coconuts knocked off their stands for different values of \(n\). \includegraphics{figure_3}

1. Describe two ways in which the distribution changes as \(n\) increases. [2]

9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.

1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.

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.

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games, [3]
2. fewer than half of the games. [2]

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

1. Calculate the mean and variance for the number of these 60 games that Bhim loses. [2]
2. Using a suitable approximation calculate the probability that Bhim loses more than 4 games. [3]

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.

3 An auction house offers items of jewellery for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewellery offered for sale.

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.

1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)

When more than one game is played the games are independent.
Sam plays 20 games.

1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
5. Find, in terms of \(r\), Tessa's expected profit.

3 Visitors to a Wildlife Park in Africa have independent probabilities of 0.9 of seeing giraffes, 0.95 of seeing elephants, 0.85 of seeing zebras and 0.1 of seeing lions.

1. Find the probability that a visitor to the Wildlife Park sees all these animals.
2. Find the probability that, out of 12 randomly chosen visitors, fewer than 3 see lions.
3. 50 people independently visit the Wildlife Park. Find the mean and variance of the number of these people who see zebras.

3 Visitors to a Wildlife Park in Africa have independent probabilities of 0.9 of seeing giraffes, 0.95 of seeing elephants, 0.85 of seeing zebras and 0.1 of seeing lions.

1. Find the probability that a visitor to the Wildlife Park sees all these animals.
2. Find the probability that, out of 12 randomly chosen visitors, fewer than 3 see lions.
3. 50 people independently visit the Wildlife Park. Find the mean and variance of the number of these people who see zebras.

7 Each day Annabel eats rice, potato or pasta. Independently of each other, the probability that she eats rice is 0.75 , the probability that she eats potato is 0.15 and the probability that she eats pasta is 0.1 .

1. Find the probability that, in any week of 7 days, Annabel eats pasta on exactly 2 days.
2. Find the probability that, in a period of 5 days, Annabel eats rice on 2 days, potato on 1 day and pasta on 2 days.
3. Find the probability that Annabel eats potato on more than 44 days in a year of 365 days.

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 \(n\).

3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.

1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).

1. A fair coin is tossed 4 times.

Find the probability that

1. an equal number of head and tails occur
2. all the outcomes are the same,
3. the first tail occurs on the third throw.

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.

3 A random variable \(X\) has the distribution \(\mathrm { B } ( 13,0.12 )\).

1. Find \(\mathrm { P } ( X < 2 )\). Two independent values of \(X\) are found.
2. Find the probability that exactly one of these values is equal to 2 .

1 A fair coin is tossed 5 times and the number of heads is recorded.

1. The random variable \(X\) is the number of heads. State the mean and variance of \(X\).
2. The number of heads is doubled and denoted by the random variable \(Y\). State the mean and variance of \(Y\).

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 biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)

1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures.

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.

Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that

1. none of the dice show a score of 6, [3 marks]
2. more than one of the dice shows a score of 6, [4 marks]
3. there are equal numbers of odd and even scores showing on the dice. [3 marks]

One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]

2.The discrete random variable \(X\) follows the binomial distribution $$X \sim \mathrm {~B} ( n , p )$$ where \(0 < p < 1\) .The mode of \(X\) is \(m\) .

1. Write down,in terms of \(m , n\) and \(p\) ,an expression for \(\mathrm { P } ( X = m )\)
2. Determine,in terms of \(n\) and \(p\) ,an interval of width 1 ,in which \(m\) lies.
3. Find a value of \(n\) where \(n > 100\) ,and a value of \(p\) where \(p < 0.2\) ,for which \(X\) has two modes. For your chosen values of \(n\) and \(p\) ,state these two modes.

11 The random variable \(X\) takes the value 1 with probability \(p\) and the value 0 with probability \(1 - p\).

1. Find each of the following.
   Use the results of part (a) to prove that