5.02k Calculate Poisson probabilities

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 .

5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .

1. Show that the variance of the number of faulty screws in a packet is 1.104 .
2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.

2 The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 25.2 )\).

1. State two assumptions that are required for the Poisson model to be valid in this context.
2. 1. Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.
   2. Find the probability that, in two randomly chosen 1 -hour periods, exactly 1 order will arrive in one of the 1 -hour periods, and at least 2 orders will arrive in the other 1 -hour period. [4]
3. The shop can only deal with a maximum of 120 orders during any 36-hour period. Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.

5 Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per match. In their away matches they score goals at a constant average rate of 0.8 goals per match. You may assume that goals are scored at random times and independently of one another.

1. A week is chosen at random.
   1. Find the probability that the team scores a total of 4 goals in their two matches.
   2. Find the probability that the team scores a total of 4 goals, with more goals scored in the home match than in the away match.
2. Use a suitable approximating distribution to find the probability that the team scores fewer than 25 goals in 10 randomly chosen weeks.
3. Justify the use of the approximating distribution used in part (b).

3 In the data-entry department of a certain firm, it is known that \(0.12 \%\) of data items are entered incorrectly, and that these errors occur randomly and independently.

1. A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by \(X\).
   1. State the distribution of \(X\), including the values of any parameters.
   2. State an appropriate approximating distribution for \(X\), including the values of any parameters. Justify your choice of approximating distribution.
   3. Use your approximating distribution to find \(\mathrm { P } ( X > 2 )\).
2. Another large random sample of \(n\) data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 . Use an approximating distribution to find the largest possible value of \(n\).

5

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\).
   1. State the values that \(X\) can take.
      It is given that \(\mathrm { P } ( X = 1 ) = 3 \times \mathrm { P } ( X = 0 )\).
   2. Find \(\lambda\).
   3. Find \(\mathrm { P } ( 4 \leqslant X \leqslant 6 )\).
2. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\) where \(\mu\) is large. Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y < 46 ) = 0.0668\), correct to 4 decimal places. Find \(\mu\).

6 Between 7 p.m. and 11 p.m., arrivals of patients at the casualty department of a hospital occur at random at an average rate of 6 per hour.

1. Find the probability that, during any period of one hour between 7 p.m. and 11 p.m., exactly 5 people will arrive.
2. A patient arrives at exactly 10.15 p.m. Find the probability that at least one more patient arrives before 10.35 p.m.
3. Use a suitable approximation to estimate the probability that fewer than 20 patients arrive at the casualty department between 7 p.m. and 11 p.m. on any particular night.

6 Computer breakdowns occur randomly on average once every 48 hours of use.

1. Calculate the probability that there will be fewer than 4 breakdowns in 60 hours of use.
2. Find the probability that the number of breakdowns in one year (8760 hours) of use is more than 200.
3. Independently of the computer breaking down, the computer operator receives phone calls randomly on average twice in every 24 -hour period. Find the probability that the total number of phone calls and computer breakdowns in a 60-hour period is exactly 4 .

4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).

1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.

2 The random variable \(X\) has the distribution \(\mathrm { B } ( 400,0.01 )\).

1. Find \(\operatorname { Var } ( 4 X + 2 )\).
2. 1. State an appropriate approximating distribution for \(X\), giving the values of any parameters. Justify your choice of approximating distribution.
   2. Use your approximating distribution to find \(\mathrm { P } ( 2 \leqslant X \leqslant 5 )\).

7 Customers arrive at a particular shop at random times. It has been found that the mean number of customers who arrive during a 5 -minute interval is 2.1 .

1. Find the probability that exactly 4 customers arrive during a 10 -minute interval.
2. Find the probability that at least 4 customers arrive during a 20 -minute interval.
3. Use a suitable approximating distribution to find the probability that fewer than 40 customers arrive during a 2-hour interval.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable \(X\), which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of \(\\) 5$ to the team for each goal that they score in the next 10 matches. Find the expectation and standard deviation of the amount that the supporter will pay.

3 The local council claims that the average number of accidents per year on a particular road is 0.8 . Jane claims that the true average is greater than 0.8 . She looks at the records for a random sample of 3 recent years and finds that the total number of accidents during those 3 years was 5 .

1. Assume that the number of accidents per year follows a Poisson distribution.
   1. State null and alternative hypotheses for a test of Jane's claim.
   2. Test at the \(5 \%\) significance level whether Jane's claim is justified.
2. Jane finds that the number of accidents per year has been gradually increasing over recent years. State how this might affect the validity of the test carried out in part (a)(ii).

5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).

1. State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
   Use your approximating distribution to answer parts (b) and (c).
2. Find \(\mathrm { P } ( X \leqslant 3 )\).
3. Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
   The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\).
4. Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).

5 Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.

1. Find the probability that more than 4 cars arrive during a 20-minute period.
2. Use an approximating distribution to find the probability that the number of cars that arrive during a 12-hour period is between 150 and 160 inclusive.
   Independently of cars, trucks arrive at the fuel station at random and at a constant average rate of 3.6 per 15-minute period.
3. Find the probability that the total number of cars and trucks arriving at the fuel station during a 10 -minute period is more than 3 and less than 7 .

4 The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and council officer wishes to test at the \(2 \%\) significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2 -minute period.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test.
   The officer notes that, during a randomly chosen 2 -minute period on a weekday morning, exactly 5 cars arrive at the junction.
3. Carry out the test.
4. State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).
   The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean 4.6 per minute.
5. Use a suitable approximating distribution to find the probability that more than 300 cars will arrive at this junction in an hour.

5 The number of clients who arrive at an information desk has a Poisson distribution with mean 2.2 per 5-minute period.

1. Find the probability that, in a randomly chosen 15 -minute period, exactly 6 clients arrive at the desk.
2. If more than 4 clients arrive during a 5 -minute period, they cannot all be served. Find the probability that, during a randomly chosen 5 -minute period, not all the clients who arrive at the desk can be served.
3. Use a suitable approximating distribution to find the probability that, during a randomly chosen 1-hour period, fewer than 20 clients arrive at the desk.

7 The number of accidents per week at a certain factory has a Poisson distribution. In the past the mean has been 1.9 accidents per week. Last year, the manager gave all his employees a new booklet on safety. He decides to test, at the 5\% significance level, whether the mean number of accidents has been reduced. He notes the number of accidents during 4 randomly chosen weeks this year.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test and state the probability of a Type I error.
3. State what is meant by a Type I error in this context.
4. During the 4 randomly chosen weeks there are a total of 3 accidents. State the conclusion that the manager should reach. Give a reason for your answer.
5. Assuming that the mean remains 1.9 accidents per week, use a suitable approximation to calculate the probability that there will be more than 100 accidents during a 52-week period.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The number, \(X\), of books received at a charity shop has a constant mean of 5.1 per day.

1. State, in context, one condition for \(X\) to be modelled by a Poisson distribution.
   Assume now that \(X\) can be modelled by a Poisson distribution.
2. Find the probability that exactly 10 books are received in a 3-day period.
3. Use a suitable approximating distribution to find the probability that more than 180 books are received in a 30-day period.
   The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean 2.5 per day.
4. Find the probability that the total number of books and DVDs that are received at the shop in 1 day is more than 3 .

8 A new light was installed on a certain footpath. A town councillor decided to use a hypothesis test to investigate whether the number of people using the path in the evening had increased. Before the light was installed, the mean number of people using the path during any 20 -minute period during the evening was 1.01. After the light was installed, the total number, \(n\), of people using the path during 3 randomly chosen 20 -minute periods during the evening was noted.

1. Given that the value of \(n\) was 6 , use a Poisson distribution to carry out the test at the \(5 \%\) significance level.
2. Later a similar test, at the \(5 \%\) significance level, was carried out using another 3 randomly chosen 20 -minute periods during the evening. Find the probability of a Type I error.
3. State what is meant by a Type I error in this context.
4. State, in context, what further information would be needed in order to find the probability of a Type II error. Do not carry out any further calculation.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Sales of cell phones at a certain shop occur singly, randomly and independently.

1. State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.
   The average number of sales per hour is 1.2 .
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .
3. Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1-month period (140 hours) will be less than 150 .

4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).

1. State the distribution of \(X\), giving the values of any parameters.
2. State an approximating distribution for \(X\), giving the values of any parameters.
3. Use this approximating distribution to find each of the following.
   1. \(\mathrm { P } ( X = 4 )\).
   2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.

7

1. Two ponds, \(A\) and \(B\), each contain a large number of fish. It is known that \(2.4 \%\) of fish in pond \(A\) are carp and \(1.8 \%\) of fish in pond \(B\) are carp. Random samples of 50 fish from pond \(A\) and 60 fish from pond \(B\) are selected. Use appropriate Poisson approximations to find the following probabilities.
   1. The samples contain at least 2 carp from pond \(A\) and at least 2 carp from pond \(B\).
   2. The samples contain at least 4 carp altogether.
2. The random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. It is given that
   Find the value of \(k\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4

1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).

6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that

1. exactly 3 people arrive at the checkout during a 5 -minute period,
2. at least 30 people arrive at the checkout during a 1-hour period. People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.
3. Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.