Two independent Poisson sums

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).

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 .

5 The number of goals scored by a sports team in the first half of any match has the distribution \(X \sim \mathrm { Po }\) (3.1). The number of goals scored by the same team in the second half of any match has the distribution \(Y \sim \operatorname { Po } ( 2.4 )\). You may assume that the distributions of \(X\) and \(Y\) are independent.

1. Find \(\mathrm { P } ( X < 4 )\).
2. Find the probability that, in a randomly chosen match, the team scores at least 5 goals.
3. Given that the team scores a total of 5 goals in a randomly chosen match, find the probability that they score exactly 3 goals in the first half.

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.

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.

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]{4f215475-30fd-47fb-aa77-0b53e339f50c-08_2720_35_109_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-09_2716_29_107_22}
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 lions seen per day during a standard safari has the distribution \(\operatorname { Po } ( 0.8 )\). The number of lions seen per day during an off-road safari has the distribution \(\operatorname { Po } ( 2.7 )\). The two distributions are independent.

1. Susan goes on a standard safari for one day. Find the probability that she sees at least 2 lions.
2. Deena goes on a standard safari for 3 days and then on an off-road safari for 2 days. Find the probability that she sees a total of fewer than 5 lions.
3. Khaled goes on a standard safari for \(n\) days, where \(n\) is an integer. He wants to ensure that his chance of not seeing any lions is less than \(10 \%\). Find the smallest possible value of \(n\).

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.

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 .

1 The number of radioactive particles emitted per second by a certain metal is random and has mean 1.7. The radioactive metal is placed next to an object which independently emits particles at random such that the mean number of particles emitted per second is 0.6 . Find the probability that the total number of particles emitted in the next 3 seconds is 6, 7 or 8 .

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.

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 .

1 A car repair firm receives call-outs both as a result of breakdowns and also as a result of accidents. On weekdays (Monday to Friday), call-outs resulting from breakdowns occur at random, at an average rate of 6 per 5 -day week; call-outs resulting from accidents occur at random, at an average rate of 2 per 5 -day week. The two types of call-out occur independently of each other. Find the probability that the total number of call-outs received by the firm on one randomly chosen weekday is more than 3 .

1 The numbers of minor flaws that occur on reels of copper wire and reels of steel wire have Poisson distributions with means 0.21 per metre and 0.24 per metre respectively. One length of 5 m is cut from each reel.

1. Calculate the probability that the total number of flaws on these two lengths of wire is at least 2 .
2. State one assumption needed in the calculation.

1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.

2 The random variable \(A\) has a Poisson distribution with mean 2 The random variable \(B\) has a Poisson distribution with standard deviation 4 The random variables \(A\) and \(B\) are independent.
State the distribution of \(A + B\) Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)

4 The number of printers, \(V\), bought during one day from the Verigood store can be modelled by a Poisson distribution with mean 4.5 The number of printers, \(W\), bought during one day from the Winnerprint store can be modelled by a Poisson distribution with mean 5.5 4

1. Find the probability that the total number of printers bought during one day from Verigood and Winnerprint stores is greater than 10.
   [0pt] [2 marks] 4
2. State the circumstance under which the distributional model you used in part (a) would not be valid.
   [0pt] [1 mark]

2 A new information technology centre is advertising places on its one-week residential computer courses.

1. The number of places, \(X\), booked each week on the publishing course may be modelled by a Poisson distribution with a mean of 9.0.
   1. State the standard deviation of \(X\).
   2. Calculate \(\mathrm { P } ( 6 < X < 12 )\).
2. The number of places booked each week on the web design course may be modelled by a Poisson distribution with a mean of 2.5.
   1. Write down the distribution for \(T\), the total number of places booked each week on the publishing and web design courses.
   2. Hence calculate the probability that, during a given week, a total of fewer than 2 places are booked.
3. The number of places booked on the database course during each of a random sample of 10 weeks is as follows: $$\begin{array} { l l l l l l l l l l } 14 & 15 & 8 & 16 & 18 & 4 & 10 & 12 & 15 & 8 \end{array}$$ By calculating appropriate numerical measures, state, with a reason, whether or not the Poisson distribution \(\mathrm { Po } ( 12.0 )\) could provide a suitable model for the number of places booked each week on the database course.

2

1. The number of telephone calls, \(X\), received per hour for Dr Able may be modelled by a Poisson distribution with mean 6 . Determine \(\mathrm { P } ( X = 8 )\).
2. The number of telephone calls, \(Y\), received per hour for Dr Bracken may be modelled by a Poisson distribution with mean \(\lambda\) and standard deviation 3 .
   1. Write down the value of \(\lambda\).
   2. Determine \(\mathrm { P } ( Y > \lambda )\).
3. 1. Assuming that \(X\) and \(Y\) are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken.
   2. Determine the probability that a total of at most 20 telephone calls will be received during any one-hour period.
   3. The total number of telephone calls received during each of 6 one-hour periods is to be recorded. Calculate the probability that a total of at least 21 telephone calls will be received during exactly 4 of these one-hour periods.

3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, \(X\), for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, \(Y\), for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.

1. Calculate:
   1. \(\mathrm { P } ( X \leqslant 3 )\);
   2. \(\quad \mathrm { P } ( Y = 5 )\).
2. The total daily demand for the device at Joe's two garages is denoted by \(T\).
   1. Write down the distribution of \(T\), stating any assumption that you make.
   2. Determine \(\mathrm { P } ( 6 < T < 12 )\).
   3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
   4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least \(99 \%\) of days.

2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.

1. For any particular day, find:
   1. \(\mathrm { P } ( X = 3 )\);
   2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
2. 1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
   2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
   3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)

2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

1. Find, for a given weekday:
   1. \(\mathrm { P } ( X < 4 )\);
   2. \(\quad \mathrm { P } ( Y = 4 )\).
2. 1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
   2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
   1. Calculate the mean and the variance of these data.
   2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.