Poisson with binomial combination

6 A dressmaker makes dresses for Easifit Fashions. Each dress requires \(2.5 \mathrm {~m} ^ { 2 }\) of material. Faults occur randomly in the material at an average rate of 4.8 per \(20 \mathrm {~m} ^ { 2 }\).

1. Find the probability that a randomly chosen dress contains at least 2 faults. Each dress has a belt attached to it to make an outfit. Independently of faults in the material, the probability that a belt is faulty is 0.03 . Find the probability that, in an outfit,
2. neither the dress nor its belt is faulty,
3. the dress has at least one fault and its belt is faulty. The dressmaker attaches 300 randomly chosen belts to 300 randomly chosen dresses. An outfit in which the dress has at least one fault and its belt is faulty is rejected.
4. Use a suitable approximation to find the probability that fewer than 3 outfits are rejected.

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.

2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

1. Explain the meaning of the term 'independently' and name the distribution that models this situation.
2. Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
3. Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
4. A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
5. State the exact distribution of the number of boxes which have at least 8 defects.
6. Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.

5. A school photocopier breaks down randomly at a rate of 15 times per year.

1. Find the probability that there will be exactly 3 breakdowns in the next month.
2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
4. Find the probability that the head teacher will buy a new photocopier.

1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)

The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).

1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
2. 1. Write down suitable hypotheses for this test.
   2. Describe a suitable test statistic that the manager should use.
   3. Explain what is meant by the critical region for this test.
3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
4. Find the actual significance level for this test.

1. A company produces steel cable.

Defects in the steel cable produced by this company occur at random, at a constant rate of 1 defect per 16 metres. On one day the company produces a piece of steel cable 80 metres long.

1. Find the probability that there are at most 5 defects in this piece of steel cable. The company produces a piece of steel cable 80 metres long on each of the next 4 days.
2. Find the probability that fewer than 2 of these 4 pieces of steel cable contain at most 5 defects. The following week the company produces a piece of steel cable \(x\) metres long.
   Using a normal approximation, the probability that this piece of steel cable has fewer than 26 defects is 0.5398
3. Find the value of \(x\)

7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .

1. Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
2. Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length \(x \mathrm {~m}\) is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
3. Find the value of \(x\).

1. The independent random variables \(W\) and \(X\) have the following distributions.

$$W \sim \operatorname { Po } ( 4 ) \quad X \sim \mathrm {~B} ( 3,0.8 )$$

1. Write down the value of the variance of \(W\)
2. Determine the mode of \(X\) Show your working clearly. One observation from each distribution is recorded as \(W _ { 1 }\) and \(X _ { 1 }\) respectively.
3. Find \(\mathrm { P } \left( W _ { 1 } = 2 \right.\) and \(\left. X _ { 1 } = 2 \right)\)
4. Find \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)

5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .

1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.

1. In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3

Find the probability that

1. exactly 4 customers join the queue in the next 10 minutes,
2. more than 10 customers join the queue in the next 20 minutes. When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, \(T\) minutes, has a continuous uniform distribution over the interval \([ 0,5 ]\). The random variable \(T\) is independent of the number of people joining the queue.
3. Find \(\mathrm { P } ( T > 3.5 )\) In a random sample of 5 customers, the random variable \(C\) represents the number of customers who took more than 3.5 minutes paying the assistant.
4. Find \(\mathrm { P } ( C \geqslant 3 )\) Bethan has just reached the front of the queue and starts paying the assistant.
5. Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.

1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.

1. 1. Write down the distribution of \(X\), the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
   2. Determine \(\mathrm { P } ( X = 20 )\).
   3. Determine \(\mathrm { P } ( 6 \leqslant X \leqslant 18 )\).
2. Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
3. When \(n\) cars pass the speed camera, the number of cars, \(Y\), that exceed 60 mph may be modelled by the distribution \(\mathrm { B } ( n , 0.2 )\). Given that \(n = 20\), determine \(\mathrm { P } ( Y \geqslant 5 )\).
4. Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .

1. A chocolate manufacturer places special tokens in \(2 \%\) of the bars it produces so that each bar contains at most one token. Anyone who collects 3 of these tokens can claim a prize.

Andreia buys a box of 40 bars of the chocolate.

1. Find the probability that Andreia can claim a prize. Barney intends to buy bars of the chocolate, one at a time, until he can claim a prize.
2. Find the probability that Barney can claim a prize when he buys his 40th bar of chocolate.
3. Find the expected number of bars that Barney must buy to claim a prize.

A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution. The fisherman takes 5 fishing trips each lasting 1 hour.

1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]

The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,

1. carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]

From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m.

1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. [2]

The twine is usually sold in balls of length 100 m. A customer buys three balls of twine.

1. Find the probability that only one of them will have fewer than 6 faults. [6]

As a special order a ball of twine containing 500 m is produced.

1. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive. [6]

In a survey of 22 families, the number of children, \(X\), in each family was given by the following table, where \(f\) denotes the frequency:

\(X\)	0	1	2	3	4	5
\(f\)	3	8	5	3	2	1

1. Find the mean and variance of \(X\). [4 marks]
2. Explain why these results suggest that \(X\) may follow a Poisson distribution. [1 mark]
3. State another feature of the data that suggests a Poisson distribution. [1 mark]

It is sometimes suggested that the number of children in a family follows a Poisson distribution with mean 2·4. Assuming that this is correct,

1. find the probability that a family has less than two children. [3 marks]
2. Use this result to find the probability that, in a random sample of 22 families, exactly 11 of the families have less than two children. [3 marks]

An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.

1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
2. Find the probability that on one particular day he repairs
   1. no CD players,
   2. more than 6 CD players. [3 marks]
3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]

The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives

1. 3 complaints, [3 marks]
2. 10 or more complaints. [2 marks]

The supermarket is open on six days each week.

1. Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]

Naomi produces oak tabletops, each of area 4·8 m². Defects in the oak tabletops occur randomly at a rate of 0·25 per m².

1. Find the probability that a randomly chosen tabletop will contain at most 2 defects. [3]
2. Find the probability that, in a random sample of 7 tabletops, exactly 4 will contain at most 2 defects each. [3]

Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.

1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. [3]

Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.

1. Find the probability that in exactly 3 of these periods there were no calls. [2]

On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.

1. Find the probability that Indre missed exactly 1 call in each of these 2 breaks. [3]