5.02k Calculate Poisson probabilities

5 A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables \(G \sim \operatorname { Po } ( 0.10 )\) and \(B \sim \operatorname { Po } ( 0.15 )\) respectively.

1. Find the probability that during a randomly chosen 2-day period no girls arrive late.
2. Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3 .
3. It is given that the values of \(\mathrm { P } ( G = r )\) and \(\mathrm { P } ( B = r )\) for \(r \geqslant 3\) are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys.
   Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.
4. Test the teacher's claim at the \(5 \%\) significance level.

5 The number of absences per week by workers at a factory has the distribution \(\operatorname { Po } ( 2.1 )\).

1. Find the standard deviation of the number of absences per week.
2. Find the probability that the number of absences in a 2-week period is at least 2 .
3. Find the probability that the number of absences in a 3-week period is more than 4 and less than 8 .
   Following a change in working conditions, the management wished to test whether the mean number of absences has decreased. They found that, in a randomly chosen 3-week period, there were exactly 2 absences.
4. Carry out the test at the \(10 \%\) significance level.
5. State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (d).

5 Customers arrive at a shop at a constant average rate of 2.3 per minute.

1. State another condition for the number of customers arriving per minute to have a Poisson distribution.
   It is now given that the number of customers arriving per minute has the distribution \(\mathrm { Po } ( 2.3 )\).
2. Find the probability that exactly 3 customers arrive during a 1 -minute period.
3. Find the probability that more than 3 customers arrive during a 2 -minute period.
4. Five 1-minute periods are chosen at random. Find the probability that no customers arrive during exactly 2 of these 5 periods.

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

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.

6 The number of accidents per month, \(X\), at a factory has a Poisson distribution. In the past the mean has been 1.1 accidents per month. Some new machinery is introduced and the management wish to test whether the mean has increased. They note the number of accidents in a randomly chosen month and carry out a hypothesis test at the 1\% significance level.

1. Show that the critical region for the test is \(X \geqslant 5\). Given that the number of accidents is 6 , carry out the test.
   Later they carry out a similar test, also at the \(1 \%\) significance level.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. Given that the mean is now 7.0 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(n\).

1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .

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 \(n\).

5 The number of eagles seen per hour in a certain location has the distribution \(\operatorname { Po } ( 1.8 )\). The number of vultures seen per hour in the same location has the independent distribution \(\operatorname { Po } ( 2.6 )\).

1. Find the probability that, in a randomly chosen hour, at least 2 eagles are seen.
2. Find the probability that, in a randomly chosen half-hour period, the total number of eagles and vultures seen is less than 5 .
   Alex wants to be at least \(99 \%\) certain of seeing at least 1 eagle.
3. Find the minimum time for which she should watch for eagles.

4 The number of emergency telephone calls to the electricity board office in a certain area in \(t\) minutes is known to follow a Poisson distribution with mean \(\frac { 1 } { 80 } t\).

1. Find the probability that there will be at least 3 emergency telephone calls to the office in any 20-minute period.
2. The probability that no emergency telephone call is made to the office in a period of \(k\) minutes is 0.9 . Find \(k\).

6 A shopkeeper sells electric fans. The demand for fans follows a Poisson distribution with mean 3.2 per week.

1. Find the probability that the demand is exactly 2 fans in any one week.
2. The shopkeeper has 4 fans in his shop at the beginning of a week. Find the probability that this will not be enough to satisfy the demand for fans in that week.
3. Given instead that he has \(n\) fans in his shop at the beginning of a week, find, by trial and error, the least value of \(n\) for which the probability of his not being able to satisfy the demand for fans in that week is less than 0.05 .

6 The random variable \(X\) denotes the number of worms on a one metre length of a country path after heavy rain. It is given that \(X\) has a Poisson distribution.

1. For one particular path, the probability that \(X = 2\) is three times the probability that \(X = 4\). Find the probability that there are more than 3 worms on a 3.5 metre length of this path.
2. For another path the mean of \(X\) is 1.3.
   1. On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
   2. Find the probability that there are more than 1250 worms on a one kilometre length of this path.

6 In their football matches, Rovers score goals independently and at random times. Their average rate of scoring is 2.3 goals per match.

1. State the expected number of goals that Rovers will score in the first half of a match.
2. Find the probability that Rovers will not score any goals in the first half of a match but will score one or more goals in the second half of the match.
3. Football matches last for 90 minutes. In a particular match, Rovers score one goal in the first 30 minutes. Find the probability that they will score at least one further goal in the remaining 60 minutes. Independently of the number of goals scored by Rovers, the number of goals scored per football match by United has a Poisson distribution with mean 1.8.
4. Find the probability that a total of at least 3 goals will be scored in a particular match when Rovers play United.

2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.

1. Find the probability that more than 1 unwanted email arrives in a period of 5 hours.
2. Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.

3 An airline knows that some people who have bought tickets may not arrive for the flight. The airline therefore sells more tickets than the number of seats that are available. For one flight there are 210 seats available and 213 people have bought tickets. The probability of any person who has bought a ticket not arriving for the flight is \(\frac { 1 } { 50 }\).

1. By considering the number of people who do not arrive for the flight, use a suitable approximation to calculate the probability that more people will arrive than there are seats available. Independently, on another flight for which 135 people have bought tickets, the probability of any person not arriving is \(\frac { 1 } { 75 }\).
2. Calculate the probability that, for both these flights, the total number of people who do not arrive is 5 .

2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people every 3 minutes.

1. Find the probability that exactly 4 people arrive in a 5 -minute period. At another checkout in the same supermarket, people arrive randomly and independently at an average rate of 1 person each minute.
2. Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3 -minute period.

3 A book contains 40000 words. For each word, the probability that it is printed wrongly is 0.0001 and these errors occur independently. The number of words printed wrongly in the book is represented by the random variable \(X\).

1. State the exact distribution of \(X\), including the values of any parameters.
2. State an approximate distribution for \(X\), including the values of any parameters, and explain why this approximate distribution is appropriate.
3. Use this approximate distribution to find the probability that there are more than 3 words printed wrongly in the book.

7 In the past, the number of house sales completed per week by a building company has been modelled by a random variable which has the distribution \(\mathrm { Po } ( 0.8 )\). Following a publicity campaign, the builders hope that the mean number of sales per week will increase. In order to test at the \(5 \%\) significance level whether this is the case, the total number of sales during the first 3 weeks after the campaign is noted. It is assumed that a Poisson model is still appropriate.

1. Given that the total number of sales during the 3 weeks is 5 , carry out the test.
2. During the following 3 weeks the same test is carried out again, using the same significance level. Find the probability of a Type I error.
3. Explain what is meant by a Type I error in this context.
4. State what further information would be required in order to find the probability of a Type II error.

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 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.

4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.

1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .

4 Goals scored by Femchester United occur at random with a constant average of 1.2 goals per match. Goals scored against Femchester United occur independently and at random with a constant average of 0.9 goals per match.

1. Find the probability that in a randomly chosen match involving Femchester,
   1. a total of 3 goals are scored,
   2. a total of 3 goals are scored and Femchester wins. The manager promises the Femchester players a bonus if they score at least 35 goals in the next 25 matches.
   3. Find the probability that the players receive the bonus.

6 The number of accidents on a certain road has a Poisson distribution with mean 3.1 per 12-week period.

1. Find the probability that there will be exactly 4 accidents during an 18-week period. Following the building of a new junction on this road, an officer wishes to determine whether the number of accidents per week has decreased. He chooses 15 weeks at random and notes the number of accidents. If there are fewer than 3 accidents altogether he will conclude that the number of accidents per week has decreased. He assumes that a Poisson distribution still applies.
2. Find the probability of a Type I error.
3. Given that the mean number of accidents per week is now 0.1 , find the probability of a Type II error.
4. Given that there were 2 accidents during the 15 weeks, explain why it is impossible for the officer to make a Type II error.