2.04d Normal approximation to binomial

2 An airline has found that, on average, 1 in 100 passengers do not arrive for each flight, and that this occurs randomly. For one particular flight the airline always sells 403 seats. The plane only has room for 400 passengers, so the flight is overbooked if the number of passengers who do not arrive is less than 3 . Use a suitable approximation to find the probability that the flight is overbooked.

1

1. 1. A random variable \(X\) has the distribution \(\mathrm { B } ( 2540,0.001 )\). Use the Poisson approximation to the binomial distribution to find \(\mathrm { P } ( X > 1 )\).
   2. Explain why the Poisson approximation is appropriate in this case.
2. Two independent random variables, \(S\) and \(T\), have distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively. Find the mean and standard deviation of \(S + T\).

4 Small drops of two liquids, \(A\) and \(B\), are randomly and independently distributed in the air. The average numbers of drops of \(A\) and \(B\) per cubic centimetre of air are 0.25 and 0.36 respectively.

1. A sample of \(10 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Find the probability that the total number of drops of \(A\) and \(B\) in this sample is at least 4 .
2. A sample of \(100 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Use an approximating distribution to find the probability that the total number of drops of \(A\) and \(B\) in this sample is less than 60 .

1 On average, 1 in 150 components made by a certain machine are faulty. The random variable \(X\) denotes the number of faulty components in a random sample of 500 components.

1. Describe fully the distribution of \(X\).
2. State a suitable approximating distribution for \(X\), giving a justification for your choice.
3. Use your approximating distribution to find the probability that the sample will include at least 3 faulty components.

2 Cars arrive at a filling station randomly and at a constant average rate of 2.4 cars per minute.

1. Calculate the probability that fewer than 4 cars arrive in a 2 -minute period.
2. Use a suitable approximating distribution to calculate the probability that at least 140 cars arrive in a 1-hour period.

5

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
   
2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
   
3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
   1. Write down an inequality in \(n\).
   2. Hence or otherwise find the largest possible value of \(n\).

5 On average, 1 in 2500 adults has a certain medical condition.

1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).

2 An engineering test consists of 100 multiple-choice questions. Each question has 5 suggested answers, only one of which is correct. Ashok knows nothing about engineering, but he claims that his general knowledge enables him to get more questions correct than just by guessing. Ashok actually gets 27 answers correct. Use a suitable approximating distribution to test at the \(5 \%\) significance level whether his claim is justified.

6 Customers arrive at an enquiry desk at a constant average rate of 1 every 5 minutes.

1. State one condition for the number of customers arriving in a given period to be modelled by a Poisson distribution. Assume now that a Poisson distribution is a suitable model.
2. Find the probability that exactly 5 customers will arrive during a randomly chosen 30 -minute period.
3. Find the probability that fewer than 3 customers will arrive during a randomly chosen 12-minute period.
4. Find an estimate of the probability that fewer than 30 customers will arrive during a randomly chosen 2-hour period.

7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 1.6 )\).

1. The random variable \(R\) is the sum of three independent values of \(X\). Find \(\mathrm { P } ( R < 4 )\).
2. The random variable \(S\) is the sum of \(n\) independent values of \(X\). It is given that $$\mathrm { P } ( S = 4 ) = \frac { 16 } { 3 } \times \mathrm { P } ( S = 2 )$$ Find \(n\).
3. The random variable \(T\) is the sum of 40 independent values of \(X\). Find \(\mathrm { P } ( T > 75 )\).

7 The number of workers, \(X\), absent from a factory on a particular day has the distribution \(\mathrm { B } ( 80,0.01 )\).

1. Explain why it is appropriate to use a Poisson distribution as an approximating distribution for \(X\).
2. Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 . Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.
3. During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the \(2 \%\) significance level.

1 Each computer made in a factory contains 1000 components. On average, 1 in 30000 of these components is defective. Use a suitable approximate distribution to find the probability that a randomly chosen computer contains at least 1 faulty component.

2 The probability that a randomly chosen plant of a certain kind has a particular defect is 0.01 . A random sample of 150 plants is taken.

1. Use an appropriate approximating distribution to find the probability that at least 1 plant has the defect. Justify your approximating distribution. The probability that a randomly chosen plant of another kind has the defect is 0.02 . A random sample of 100 of these plants is taken.
2. Use an appropriate approximating distribution to find the probability that the total number of plants with the defect in the two samples together is more than 3 and less than 7 .

5 On average, 1 in 2500 adults has a certain medical condition.

1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).

2 The number of calls received per 5-minute period at a large call centre has a Poisson distribution with mean \(\lambda\), where \(\lambda > 30\). If more than 55 calls are received in a 5 -minute period, the call centre is overloaded. It has been found that the probability of being overloaded during a randomly chosen 5 -minute period is 0.01 . Use the normal approximation to the Poisson distribution to obtain a quadratic equation in \(\sqrt { } \lambda\) and hence find the value of \(\lambda\).

5

1. Narika has a die which is known to be biased so that the probability of throwing a 6 on any throw is \(\frac { 1 } { 100 }\). She uses an approximating distribution to calculate the probability of obtaining no 6s in 450 throws. Find the percentage error in using the approximating distribution for this calculation.
2. Johan claims that a certain six-sided die is biased so that it shows a 6 less often than it would if the die were fair. In order to test this claim, the die is thrown 25 times and it shows a 6 on only 2 throws. Test at the \(10 \%\) significance level whether Johan's claim is justified.

3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that

1. more than 3 particles are emitted during a 20 -second period,
2. more than 240 particles are emitted during a 1-hour period.

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.

6 The number of house sales per week handled by an estate agent is modelled by the distribution \(\operatorname { Po } ( 3 )\).

1. Find the probability that, in one randomly chosen week, the number of sales handled is
   1. greater than 4 ,
   2. exactly 4 .
   3. Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales.
   4. One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another.
      (a) Explain, in non-technical language, what you understand by this condition.
      (b) State another condition that is needed.

3

1. The random variable \(X\) has a \(\mathrm { B } ( 60,0.02 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( X \leqslant 2 )\).
2. The random variable \(Y\) has a \(\operatorname { Po } ( 30 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( Y \leqslant 38 )\).

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.

6 Customers arrive at a post office at a constant average rate of 0.4 per minute.

1. State an assumption needed to model the number of customers arriving in a given time interval by a Poisson distribution. Assuming that the use of a Poisson distribution is justified,
2. find the probability that more than 2 customers arrive in a randomly chosen 1 -minute interval,
3. use a suitable approximation to calculate the probability that more than 55 customers arrive in a given two-hour interval,
4. calculate the smallest time for which the probability that no customers arrive in that time is less than 0.02 , giving your answer to the nearest second.