5.02i Poisson distribution: random events model

2 The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.

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1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
   1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
   2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.

5 In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.

1. Find the probability that there are fewer than 3 typing errors in 10 randomly chosen pages.
2. Use an approximating distribution to find the probability that there are more than 50 typing errors in 200 randomly chosen pages.
   In the same document, formatting errors occur at random and at a constant mean rate of 0.3 per page.
3. Find the probability that the total number of typing and formatting errors in 20 randomly chosen pages is between 8 and 11 inclusive.

3 Drops of water fall randomly from a leaking tap at a constant average rate of 5.2 per minute.

1. Find the probability that at least 3 drops fall during a randomly chosen 30 -second period.
2. Use a suitable approximating distribution to find the probability that at least 650 drops fall during a randomly chosen 2-hour period.

3 A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.

1. Assume that the owner is correct.
   1. Find the probability that there will be at least 4 hits during a 10-minute period.
   2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.
      A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time ( 9.00 am to 9.00 pm ) is usually about twice the number of hits during the night-time ( 9.00 pm to 9.00 am ).
2. 1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.
   2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.

5 In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.31 )\). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5 , carry out the test at the \(2.5 \%\) significance level.

1

1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 25 )\).
   Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 30 )\).
2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 100 , p )\) where \(p < 0.05\). Use the Poisson approximation to the binomial distribution to write down an expression, in terms of \(p\), for \(\mathrm { P } ( Y < 3 )\).

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 The numbers of customers arriving at service desks \(A\) and \(B\) during a 10 -minute period have the independent distributions \(\operatorname { Po } ( 1.8 )\) and \(\operatorname { Po } ( 2.1 )\) respectively.

1. Find the probability that during a randomly chosen 15 -minute period more than 2 customers will arrive at \(\operatorname { desk } A\).
2. Find the probability that during a randomly chosen 5-minute period the total number of customers arriving at both desks is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-08_2720_35_109_2012}
3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90 \%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute.

7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-10_2716_40_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-11_2716_29_107_22}
3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
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1 A random variable \(X\) has the distribution \(\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)\).
Use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( X \geqslant 4 )\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-03_2716_29_107_22}

5 A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions \(\operatorname { Po } ( 3.7 )\) and \(\operatorname { Po } ( 2.6 )\) respectively. A bag of sweets is chosen at random.

1. Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5 .
2. Find the probability that the total number of lemon and orange sweets in the bag is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-06_2725_47_107_2002} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-07_2716_29_107_22} 10 bags of sweets are chosen at random.
3. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.

3 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5-minute period.

1. Find the probability of exactly 4 calls in an 8 -minute period.
2. Find the probability of at least 3 calls in a 3-minute period.
   The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.

6 At a certain airfield planes land at random times at a constant average rate of one every 10 minutes.

1. Find the probability that exactly 5 planes will land in a period of one hour.
2. Find the probability that at least 2 planes will land in a period of 16 minutes.
3. Given that 5 planes landed in an hour, calculate the conditional probability that 1 plane landed in the first half hour and 4 in the second half hour.

6 At a petrol station cars arrive independently and at random times at constant average rates of 8 cars per hour travelling east and 5 cars per hour travelling west.

1. Find the probability that, in a quarter-hour period,
   1. one or more cars travelling east and one or more cars travelling west will arrive,
   2. a total of 2 or more cars will arrive.
   3. Find the approximate probability that, in a 12 -hour period, a total of more than 175 cars will arrive.

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.

5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.

1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
2. Find the probability that exactly 8 people call in any half hour.
3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.

5 When a guitar is played regularly, a string breaks on average once every 15 months. Broken strings occur at random times and independently of each other.

1. Show that the mean number of broken strings in a 5 -year period is 4 . A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of 5 years.
2. The mean number of broken strings of the new type in a 5 -year period is denoted by \(\lambda\). Find the rejection region for a test at the \(10 \%\) significance level when the null hypothesis \(\lambda = 4\) is tested against the alternative hypothesis \(\lambda < 4\).
3. Hence calculate the probability of making a Type I error. The number of broken guitar strings of the new type, in a 5 -year period, was in fact 1 .
4. State, with a reason, whether there is evidence at the \(10 \%\) significance level that guitar strings of the new type break less frequently.

6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.

1. Find the probability that exactly two people arrive in a 1-minute period.
2. Find the probability that nobody arrives in a 15 -second period.
3. The probability that at least one person arrives in the next \(t\) minutes is 0.9 . Find the value of \(t\).

3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.

1. Find the probability that more than 3 major avalanches occur in a 3-month period.
2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.

6 In restaurant \(A\) an average of 2.2\% of tablecloths are stained and, independently, in restaurant \(B\) an average of 5.8\% of tablecloths are stained.

1. Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson approximation to find the probability that a total of more than 2 tablecloths are stained.
2. Random samples of \(n\) tablecloths are taken from each restaurant. The probability that at least one tablecloth is stained is greater than 0.99 . Find the least possible value of \(n\).

7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the \(10 \%\) significance level whether the mean white blood cell count has decreased.

1. State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
2. Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.
3. Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.

4 At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean number of breakdowns per year has been 4.8. Following some repairs, the management carry out a hypothesis test at the 5\% significance level to determine whether this mean has decreased. If there is at most 1 breakdown in the following year, they will conclude that the mean has decreased.

1. State what is meant by a Type I error in this context.
2. Find the probability of a Type I error.
3. Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.

7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).

1. State two assumptions required for the Poisson model to be valid.
2. Find the probability that
   1. at least 1 patient will arrive in a 15-minute period,
   2. fewer than 4 patients will arrive in a 10-minute period.
   3. The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.

1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .