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

3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{79f1015b-7c3d-4576-8d5b-e9fc89d8a49e-2_1266_1546_487_335}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to estimate the number of outliers in the sample.
3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
5. Find the probability that
   (A) exactly 2 of these 17 babies require special care,
   (B) more than 2 of the 17 babies require special care.
6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.

3 A fair dice is thrown 90 times. Use an appropriate approximation to find the probability that the number 1 is obtained 14 or more times.

5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution \(\operatorname { Po(0.2). }\)

1. Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.
2. Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .
3. Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.

5 The random variable \(W\) has the distribution \(\mathbf { B } ( 30 , p )\).

1. Use the exact binomial distribution to calculate \(\mathbf { P } ( W = 10 )\) when \(p = 0.4\).
2. Find the range of values of \(p\) for which you would expect that a normal distribution could be used as an approximation to the distribution of \(W\).
3. Use a normal approximation to calculate \(\mathrm { P } ( W = 10 )\) when \(p = 0.4\).

4

1. Explain briefly what is meant by a random sample. Random numbers are used to select, with replacement, a sample of size \(n\) from a population numbered 000, 001, 002, ..., 799.
2. If \(n = 6\), find the probability that exactly 4 of the selected sample have numbers less than 500 .
3. If \(n = 60\), use a suitable approximation to calculate the probability that at least 40 of the selected sample have numbers less than 500 .

5 An airline has 300 seats available on a flight to Australia. It is known from experience that on average only \(99 \%\) of those who have booked seats actually arrive to take the flight, the remaining \(1 \%\) being called 'no-shows'. The airline therefore sells more than 300 seats. If more than 300 passengers then arrive, the flight is over-booked. Assume that the number of no-show passengers can be modelled by a binomial distribution.

1. If the airline sells 303 seats, state a suitable distribution for the number of no-show passengers, and state a suitable approximation to this distribution, giving the values of any parameters. Using the distribution and approximation in part (i),
2. show that the probability that the flight is over-booked is 0.4165 , correct to 4 decimal places,
3. find the largest number of seats that can be sold for the probability that the flight is over-booked to be less than 0.2.

3 The proportion of adults in a large village who support a proposal to build a bypass is denoted by \(p\). A random sample of size 20 is selected from the adults in the village, and the members of the sample are asked whether or not they support the proposal.

1. Name the probability distribution that would be used in a hypothesis test for the value of \(p\).
2. State the properties of a random sample that explain why the distribution in part (i) is likely to be a good model. \(4 X\) is a continuous random variable.

6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter \(e\) appears \(19 \%\) of the time. A certain encoded message of 20 letters contains one letter \(e\).

1. Using an exact binomial distribution, test at the \(10 \%\) significance level whether there is evidence that the proportion of the letter \(e\) in the language from which this message is a sample is less than in German, i.e., less than \(19 \%\).
2. Give a reason why a binomial distribution might not be an appropriate model in this context.

2 A public water supply contains bacteria. Each day an analyst checks the water quality by counting the number of bacteria in a random sample of 5 ml of water. Throughout this question, you should assume that the bacteria occur randomly at a mean rate of 0.37 bacteria per 5 ml of water.

1. Use a Poisson distribution to
   (A) find the probability that a 5 ml sample contains exactly 2 bacteria,
   (B) show that the probability that a 5 ml sample contains more than 2 bacteria is 0.0064 .
2. The month of September has 30 days. Find the probability that during September there is at most one day when a 5 ml sample contains more than 2 bacteria. The daily 5 ml sample is the first stage of the quality control process. The remainder of the process is as follows.

3 A company has a fleet of identical vans. Company policy is to replace all of the tyres on a van as soon as any one of them is worn out. The random variable \(X\) represents the number of miles driven before the tyres on a van are replaced. \(X\) is Normally distributed with mean 27500 and standard deviation 4000.

1. Find \(\mathrm { P } ( X > 25000 )\).
2. 10 vans in the fleet are selected at random. Find the probability that the tyres on exactly 7 of them last for more than 25000 miles.
3. The tyres of \(99 \%\) of vans last for more than \(k\) miles. Find the value of \(k\). A tyre supplier claims that a different type of tyre will have a greater mean lifetime. A random sample of 15 vans is fitted with these tyres. For each van, the number of miles driven before the tyres are replaced is recorded. A hypothesis test is carried out to investigate the claim. You may assume that these lifetimes are also Normally distributed with standard deviation 4000.
4. Write down suitable null and alternative hypotheses for the test.
5. For the 15 vans, it is found that the mean lifetime of the tyres is 28630 miles. Carry out the test at the \(5 \%\) level.

2 The random variable \(X ( X = 1,2,3,4,5,6 )\) denotes the score when a fair six-sided die is rolled.

1. Write down the mean of \(X\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\).
2. Show that \(\mathrm { G } ( t )\), the probability generating function (pgf) of \(X\), is given by $$\mathrm { G } ( t ) = \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) }$$ The random variable \(N ( N = 0,1,2 , \ldots )\) denotes the number of heads obtained when an unbiased coin is tossed repeatedly until a tail is first obtained.
3. Show that \(\mathrm { P } ( N = r ) = \left( \frac { 1 } { 2 } \right) ^ { r + 1 }\) for \(r = 0,1,2 , \ldots\).
4. Hence show that \(\mathrm { H } ( t )\), the pgf of \(N\), is given by \(\mathrm { H } ( t ) = ( 2 - t ) ^ { - 1 }\).
5. Use \(\mathrm { H } ( t )\) to find the mean and variance of \(N\). A game consists of tossing an unbiased coin repeatedly until a tail is first obtained and, each time a head is obtained in this sequence of tosses, rolling a fair six-sided die. The die is not rolled on the first occasion that a tail is obtained and the game ends at that point. The random variable \(Q ( Q = 0,1,2 , \ldots )\) denotes the total score on all the rolls of the die. Thus, in the notation above, \(Q = X _ { 1 } + X _ { 2 } + \ldots + X _ { N }\) where the \(X _ { i }\) are independent random variables each distributed as \(X\), with \(Q = 0\) if \(N = 0\). The pgf of \(Q\) is denoted by \(\mathrm { K } ( t )\). The familiar result that the pgf of a sum of independent random variables is the product of their pgfs does not apply to \(\mathrm { K } ( t )\) because \(N\) is a random variable and not a fixed number; you should instead use without proof the result that \(\mathrm { K } ( t ) = \mathrm { H } ( \mathrm { G } ( t ) )\).
6. Show that \(\mathrm { K } ( t ) = 6 \left( 12 - t - t ^ { 2 } - \ldots - t ^ { 6 } \right) ^ { - 1 }\).
   [0pt] [Hint. \(\left. \left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + \ldots + t ^ { 5 } \right) .\right]\)
7. Use \(\mathrm { K } ( t )\) to find the mean and variance of \(Q\).
8. Using your results from parts (i), (v) and (vii), verify the result that (in the usual notation for means and variances) $$\sigma _ { Q } { } ^ { 2 } = \sigma _ { N } { } ^ { 2 } \mu _ { X } { } ^ { 2 } + \mu _ { N } \sigma _ { X } { } ^ { 2 } .$$

1. Some children are playing a game involving throwing a ball into a bucket. Each child has 3 throws and the number of times the ball lands in the bucket, \(x\), is recorded. Their results are given in the table below.

1. Find \(\bar { x }\) (1) Sandra decides to model the game by assuming that on each throw, the probability of the ball landing in the bucket is 0.4 for every child on every throw and that the throws are all independent. The random variable \(S\) represents the number of times the ball lands in the bucket for a randomly selected child.
2. Find \(\mathrm { P } ( S = 2 )\)
3. Complete the table below to show the probability distribution for \(S\).

   \(s\)	0	1	2	3
   \(\mathrm { P } ( S = s )\)		0.432		0.064

   Ting believes that the probability of the ball landing in the bucket is not the same for each throw. He suggests that the probability will increase with each throw and uses the model $$p _ { i } = 0.15 i + 0.10$$ where \(i = 1,2,3\) and \(p _ { i }\) is the probability that the \(i\) th throw of the ball, by any particular child, will land in the bucket.
   The random variable \(T\) represents the number of times the ball lands in the bucket for a randomly selected child using Ting's model.
4. Show that
   1. \(\mathrm { P } ( T = 3 ) = 0.055\)
   2. \(\mathrm { P } ( T = 1 ) = 0.45\) (5)
5. Complete the table below to show the probability distribution for \(T\), stating the exact probabilities in each case.

   \(t\)	0	1	2	3
   \(\mathrm { P } ( T = t )\)		0.45		0.055

6. State, giving your reasons, whether Sandra's model or Ting's model is the more appropriate for modelling this game.

4. A spinner can land on the numbers \(10,12,14\) and 16 only and the probability of the spinner landing on each number is the same.
The random variable \(X\) represents the number that the spinner lands on when it is spun once.

1. State the name of the probability distribution of \(X\).
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find \(\operatorname { Var } ( X )\) A second spinner can land on the numbers 1, 2, 3, 4 and 5 only. The random variable \(Y\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(Y\) is given in the table below

      \(y\)	1	2	3	4	5
      \(\mathrm { P } ( Y = y )\)	\(\frac { 4 } { 30 }\)	\(\frac { 9 } { 30 }\)	\(\frac { 6 } { 30 }\)	\(\frac { 5 } { 30 }\)	\(\frac { 6 } { 30 }\)

3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\) The random variable \(W = a X + b\), where \(a\) and \(b\) are constants and \(a > 0\) Given that \(\mathrm { E } ( W ) = \mathrm { E } ( Y )\) and \(\operatorname { Var } ( W ) = \operatorname { Var } ( Y )\)
4. find the value of \(a\) and the value of \(b\). Each of the two spinners is spun once.
5. Find \(\mathrm { P } ( W = Y )\)

1. A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.

If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won. By considering the possible positions for the centre of the disc,

1. show that the probability of winning a prize on any particular roll is \(\frac { 1 } { 5 }\) A group of 15 students each roll the disc onto the grid twenty times and record the number of times, \(x\), that each student wins a prize. Their results are summarised as follows $$\sum x = 61 \quad \sum x ^ { 2 } = 295$$
2. Find the standard deviation of the number of prizes won per student. A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2
3. Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students. The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).
4. Explain how this probability could be used to find an estimate for the value of \(\pi\) and state the value of your estimate.

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.

5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.

1. Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
2. Justify your use of the approximation in part (a).
   On another occasion, the same die is thrown repeatedly until a 3 is obtained.
3. Find the probability that obtaining a 3 requires fewer than 7 throws.
   

8

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 30,0.6 )\). Find \(\mathrm { P } ( X \geqslant 16 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 4,0.7 )\).
   1. Find \(\mathrm { P } ( Y = 2 )\).
   2. Three values of \(Y\) are chosen at random. Find the probability that their total is 10 .

4 Each time Ben attempts to complete a crossword in his daily newspaper, the probability that he succeeds is \(\frac { 2 } { 3 }\). The random variable \(X\) denotes the number of times that Ben succeeds in 9 attempts.

1. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( X < 6 )\),
   3. \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). Ben notes three values, \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), of \(X\).
   4. State the total number of attempts to complete a crossword that are needed to obtain three values of \(X\). Hence find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } = 18 \right)\).

5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .

1. Find the probability that the first time Jack wins a ticket is
   1. on his 8th attempt,
   2. after his 8th attempt.
   3. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
   4. Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.

7 Froox sweets are packed into tubes of 10 sweets, chosen at random. \(25 \%\) of Froox sweets are yellow.

1. Find the probability that in a randomly selected tube of Froox sweets there are
   1. exactly 3 yellow sweets,
   2. at least 3 yellow sweets.
   3. Find the probability that in a box containing 6 tubes of Froox sweets, there is at least 1 tube that contains at least 3 yellow sweets.

6 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.

1. (A) Find the probability that there are exactly 4 smokers in the sample.
   (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample.
   (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Explain why the alternative hypothesis has the form that it does.
3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
4. In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.

7 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.

1. 10 customers are selected at random.
   (A) Find the probability that exactly 5 of them are accessing the internet.
   (B) Find the probability that at least 5 of them are accessing the internet.
   (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.

8 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.

1. 15 children are selected at random.
   (A) Find the probability that exactly 3 of them eat at least five a day.
   (B) Find the probability that at least 3 of them eat at least five a day.
   (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Give a reason for your choice of the alternative hypothesis.
3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.

6 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.

1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
   (B) Find the probability that there are more than 2 faulty tiles in the sample.
   (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Explain why the alternative hypothesis has the form that it does.
3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.

3 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.