Scaled time period sums

1 Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.1 at factory \(A\) and 1.7 at factory \(B\). Find the probability that the total number of accidents in the two factories during a \(2 -\) month period is more than 3 .

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 .

4 Bacteria of a certain type are randomly distributed in the water in two ponds, \(A\) and \(B\). The average numbers of bacteria per \(\mathrm { cm } ^ { 3 }\) in \(A\) and \(B\) are 0.32 and 0.45 respectively.

1. Samples of \(8 \mathrm {~cm} ^ { 3 }\) of water from \(A\) and \(12 \mathrm {~cm} ^ { 3 }\) of water from \(B\) are taken at random. Find the probability that the total number of bacteria in these samples is at least 3 .
2. Find the probability that in a random sample of \(155 \mathrm {~cm} ^ { 3 }\) of water from \(A\), the number of bacteria is less than 35 .

1 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.

4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.

1. Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.
2. Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.

5. 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. more than 5 . Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
3. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. 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.
4. 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.

10. On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10 -minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).

4. The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematical examinations each year can be modelled by a Poisson distribution with a mean of 3 .
a. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A-grades in their Mathematics examinations.
b. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with mean of 7 .
i. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A-grades in their Mathematics and English examinations.
ii. What assumption did you make in answering part (b)(i)?
[0pt] [Total 7 marks]

6. The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematical examinations each year can be modelled by a Poisson distribution with a mean of 3 .
a. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations.
b. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with mean of 7 . Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A-grades in their Mathematics and English examinations.
c. Lowkey School is given a performance rating, \(P = 2 X + 3 Y\), based on the number of A-grades achieved in Mathematics and English. Find: $$\begin{array} { l l } \text { i. } & \mathrm { E } ( P )
\text { ii. } & \operatorname { Var } ( P ) \end{array}$$ d. What assumption did you make in answering part (b)? Did you need this assumption to answer part (c)? Justify your answers.