5.02i Poisson distribution: random events model

1.6% of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.

1. Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle. [4]
2. Justify your approximating distribution. [2]

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. [3]
   2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period. [4]

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 (9.00am to 9.00pm) is usually about twice the number of hits during the night-time (9.00pm to 9.00am).

1. 1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution. [1]
   2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time. [2]

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

1. Find the probability that during a randomly chosen \(15\)-minute period more than \(2\) customers will arrive at desk \(A\). [2]
2. Find the probability that during a randomly chosen \(5\)-minute period the total number of customers arriving at both desks is less than \(4\). [3]
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. [4]

The number of goals scored per match by Everly Rovers is represented by the random variable \(X\) which has mean 1.8.

1. State two conditions for \(X\) to be modelled by a Poisson distribution. [2]

Assume now that \(X \sim \text{Po}(1.8)\).

1. Find \(\text{P}(2 < X < 6)\). [2]
2. The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus. [3]

The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\text{Po}(6)\). The number of child customers arriving in the same shop during a 10-minute period is modelled by an independent random variable with distribution \(\text{Po}(4.5)\).

1. Find the probability that during a randomly chosen 2-minute period, the total number of adult and child customers who arrive in the shop is less than 3. [3]
2. During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30-minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level. [6]

1\% of adults in a certain country own a yellow car.

1. Use a suitable approximating distribution to find the probability that a random sample of 240 adults includes more than 2 who own a yellow car. [4]
2. Justify your approximation. [2]

1.5% of the population of the UK can be classified as 'very tall'.

1. The random variable \(X\) denotes the number of people in a sample of \(n\) people who are classified as very tall. Given that E\((X) = 2.55\), find \(n\). [2]
2. By using the Poisson distribution as an approximation to a binomial distribution, calculate an approximate value for the probability that a sample of size 210 will contain fewer than 3 people who are classified as very tall. [3]

The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.

1. A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
2. Find the probability of a Type I error. [2]

\(X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.

1. Find P\((X + Y > 3)\). [4]
2. A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4. [4]

The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.

1. Find the probability that, in a half-hour period,
   1. 2 or more men and 1 or more women will visit the clinic, [4]
   2. a total of 3 or more people will visit the clinic. [3]
2. Find the probability that, in a 10-hour period, a total of more than 60 people will visit the clinic. [4]

The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.

1. Find the probability exactly 4 calls in a 10 minute period. [2]
2. Find the probability at least 3 calls in a 3 minute period. [3]

The number of calls received at a large call centre has a Poisson distribution with mean 4 calls per 5 minute period.

1. [(c)] Use an approximation to find the probability that the number of calls received in a 5 minute period is between 4 and 9 inclusive. [4]

Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.

1. Explain how this information tends to support the manager's suspicion. [2]

The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.

Number of breakages	0	1	2	3	4	5	6	7 or more
Frequency	21	33	46	31	23	16	10	0

The mean number of breakages calculated from this sample is 2.5.

1. Use the data from this larger sample to carry out a goodness of fit test, at the 10% significance level, to test the claim. [8]

Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.

1. 1. Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
   2. Evaluate your expression, giving your answer to 3 significant figures. [3]
2. Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
3. Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
4. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]

The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1

1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]

The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.

1. Use the tables to find the value of \(w\) [3]

A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.

1. State the null hypothesis for this test. [1]
2. Determine the critical region for the test at the 5\% level of significance. [2]

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]

On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407, to 3 significant figures. [1] Find the probability of
2. exactly 2 accidents in the next 6 month period, [3]
3. no accidents in exactly 2 of the next 4 months. [3]

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]
2. more than 5. [2]

Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.

1. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. [3]

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.

1. 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]

The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the new salesman has increased house sales. [7]

An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.

1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour. [2]

Find the probability that in a randomly chosen hour

1. all Internet users connect at their first attempt, [2]
2. more than 4 users fail to connect at their first attempt. [2]

1. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period. [1]
2. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period. [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]

A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.

1. Write down two conditions that must apply for this model to be applicable. [2]

Assuming this model and a mean of 0.7 weeds per m², find

1. the probability that in a randomly chosen plot of size 4 m² there will be fewer than 3 of these weeds, [4]
2. Using a suitable approximation, find the probability that in a plot of 100 m² there will be more than 66 of these weeds. [6]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable Y ~ Po(30).

1. Estimate P(Y > 28). [6]

A doctor expects to see, on average, 1 patient per week with a particular disease.

1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]

The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]

Medical research into the nature of the disease discovers that it can be passed from one patient to another.

1. Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]

Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.

1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. [2]

Find the probability that in any randomly selected 10 minute interval

1. exactly 6 cars pass this point, [3]
2. at least 9 cars pass this point. [2]

After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.

1. Test, at the 5\% level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly. [6]