Single time period probability

5 Eggs are sold in boxes of 20. Cracked eggs occur independently and the mean number of cracked eggs in a box is 1.4 .

1. Calculate the probability that a randomly chosen box contains exactly 2 cracked eggs.
2. Calculate the probability that a randomly chosen box contains at least 1 cracked egg.
3. A shop sells \(n\) of these boxes of eggs. Find the smallest value of \(n\) such that the probability of there being at least 1 cracked egg in each box sold is less than 0.01 .

7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 2.4 )\).

1. Find \(\mathrm { P } ( 2 \leqslant X < 4 )\).
2. Two independent values of \(X\) are chosen. Find the probability that both of these values are greater than 1 .
3. It is given that \(\mathrm { P } ( X = r ) < \mathrm { P } ( X = r + 1 )\).
   1. Find the set of possible values of \(r\).
   2. Hence find the value of \(r\) for which \(\mathrm { P } ( X = r )\) is greatest.
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6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first draft there are

1. exactly 2 errors in 10 pages,
2. at least 3 errors in 6 pages,
3. fewer than 50 errors in 200 pages.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 2.3 )\). Find \(\mathrm { P } ( 2 \leq X < 5 )\).

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1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).

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.
   • If the 5 ml sample contains more than 2 bacteria, then a 50 ml sample is taken.
   • If this 50 ml sample contains more than 8 bacteria, then a sample of 1000 ml is taken.
   • If this 1000 ml sample contains more than 90 bacteria, then the supply is declared to be 'questionable'.
   • Find the probability that a random sample of 50 ml contains more than 8 bacteria.
   • Use a suitable approximating distribution to find the probability that a random sample of 1000 ml contains more than 90 bacteria.
   • Find the probability that the supply is declared to be questionable.

1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.

The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75

1. Find the largest possible value of \(v\)
2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
   

2. The random variable \(J\) has a Poisson distribution with mean 4.

1. Find \(\mathrm { P } ( J \geqslant 10 )\). The random variable \(K\) has a binomial distribution with parameters \(n = 25 , p = 0.27\).
2. Find \(\mathrm { P } ( K \leqslant 1 )\).

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 defects per metre in a roll of cloth has a Poisson distribution with mean 0.25

Find the probability that

1. a randomly chosen metre of cloth has 1 defect,
2. the total number of defects in a randomly chosen 6 metre length of cloth is more than 2 A tailor buys 300 metres of cloth.
3. Using a suitable approximation find the probability that the tailor's cloth will contain less than 90 defects.

6. A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25 . The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. Calculate the probability that a randomly selected sample square contains
2. no sheep,
3. more than 2 sheep. A sheepdog has been sent into the field to round up the sheep.
4. Explain why the model may no longer be applicable. In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.
5. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep.

5 The number of telephone calls received, during an 8-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of 7 .

1. Determine the probability that, during a given 8 -hour period, the number of telephone calls received that request an urgent visit by an engineer is:
   1. at most 5 ;
   2. exactly 7 ;
   3. at least 5 but fewer than 10 .
2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer.
3. The IT company has 4 engineers available for urgent visits and it may be assumed that each of these engineers takes exactly 1 hour for each such visit. At 10 am on a particular day, all 4 engineers are available for urgent visits.
   1. State the maximum possible number of telephone calls received between 10 am and 11 am that request an urgent visit and for which an engineer is immediately available.
      (1 mark)
   2. Calculate the probability that at 11 am an engineer will not be immediately available to make an urgent visit.
4. Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer.
   (1 mark)
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\begin{enumerate} \item An insurance company is investigating how often its customers crash their cars.

1. Suggest an appropriate sampling frame.
2. Describe the sampling units.
3. State the advantage of a sample survey over a census in this case. \item A searchlight is rotating in a horizontal circle. It is assumed that that, at any moment, the centre of its beam is equally likely to be pointing in any direction. The random variable \(X\) represents this direction, expressed as a bearing in the range \(000 ^ { \circ }\) to \(360 ^ { \circ }\).

6. A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.

1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed.
2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning.
3. Find the probability that on a weekday morning the shop sells
   1. more than 4 pairs in a one-hour period,
   2. no pairs in a half-hour period,
   3. more than 4 pairs during each hour from 9 am until noon. The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop’s sales.
4. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in sales following the appearance of the advert.
   (4 marks)

1. (a) The random variable \(X\) follows a Poisson distribution with a mean of 1.4

Find \(\mathrm { P } ( X \leq 3 )\).
(b) The random variable \(Y\) follows a binomial distribution such that \(Y \sim \mathrm {~B} ( 20,0.6 )\). Find \(\mathrm { P } ( Y \leq 12 )\).
(4 marks)

1 The number of failures of a machine each week at a factory is modelled by a Poisson distribution with mean 0.45.

1. Write down the variance of the distribution.
2. Find the probability that there are exactly 2 failures in a week.
3. State a distribution which can be used to model the number of failures in a period of 4 weeks.
4. Find the probability that there are at least 2 failures in a period of 4 weeks.

3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.

1. Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
2. Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of \(t\) minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
3. Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
4. Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.

5. The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is

1. exactly 4,
2. more than 5 . Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
3. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
4. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased.
   (5)