1. Pure Mathematics 1

1.1 Quadratics

1.2 Functions

1.3 Coordinate geometry

1.4 Circular measure

1.5 Trigonometry

1.6 Series

1.7 Differentiation

1.8 Integration

3. Pure Mathematics 3

3.1 Algebra

3.2 Logarithmic and exponential functions

3.3 Trigonometry

3.4 Differentiation

3.5 Integration

3.6 Numerical solution of equations

3.7 Vectors

3.8 Differential equations

3.9 Complex numbers

4. Mechanics

4.1 Forces and equilibrium

4.2 Kinematics of motion in a straight line

4.3 Momentum

4.4 Newton's laws of motion

4.5 Energy, work and power

5. Probability & Statistics 1

5.1 Representation of data

5.2 Permutations and combinations

5.3 Probability

5.4 Discrete random variables

5.5 The normal distribution

6. Probability & Statistics 2

6.1 The Poisson distribution

6.2 Linear combinations of random variables

6.3 Continuous random variables

6.4 Sampling and estimation

6.5 Hypothesis tests

A-Level CAIE Mathematics A2 6.1 The Poisson distribution Question Bank

[Maximum number: 4]

The random variable X has the distribution B(4000,0.001).

(a)

Use a suitable approximating distribution to find $\mathrm{P}(2 \leqslant X<5)$ .

[ 3 ]

(b)

Justify your approximating distribution in this case.

[ 1 ]

[Maximum number: 5]

A random variable X has the distribution Po(145).

(a)

Use a suitable approximating distribution to calculate $\mathrm{P}(X \leqslant 150)$ .

[ 4 ]

(b)

Justify the use of your approximating distribution in this case.

[ 1 ]

[Maximum number: 4]

In a certain lottery, on average 1 in every 10000 tickets is a prize-winning ticket. An agent sells 6000 tickets.

(a)

Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.

[ 3 ]

(b)

Justify the use of your approximating distribution in this context.

[ 1 ]

[Maximum number: 8]

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

(a)

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.

[ 1 ]

(b)

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 ]

(c)

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 ]

[Maximum number: 4]

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.

(a)

Find the probability that during a randomly chosen 2-day period no girls arrive late.

[ 1 ]

(b)

It is given that the values of P(G=r) and 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.

[ 3 ]

[Maximum number: 4]

Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number X of days on which she sees it. She models the distribution of X by B(31, p), where p is the probability of seeing this kind of bird on a randomly chosen day in July.

Data from previous years suggests that p=0.3, but in 2022 Rita suspected that the value of p had been reduced. She decided to carry out a hypothesis test.

In July 2022, she saw this kind of bird on 4 days.

(a)

(i)

Use a suitable approximating distribution to find P(Y=4).

[ 3 ]

(ii)

Justify your approximating distribution in this context.

[ 1 ]

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Question 1

Question 1(a)

Question 1(b)

Question 1

Question 1(a)

Question 1(b)

Question 3

Question 3(a)

Question 3(b)

Question 5

Question 5(a)

Question 5(b)

Question 5(c)

Question 5

Question 5(a)

Question 5(c)

Question 7

Question 7(c)

Question 7(c)(i)

Question 7(c)(ii)