Conditional probability with Poisson

CAIE S2 2020 June Q4

8 marks Standard +0.3

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

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

CAIE S2 2018 June Q4

7 marks Standard +0.8

4 The numbers, $M$ and $F$, of male and female students who leave a particular school each year to study engineering have means 3.1 and 0.8 respectively.

State, in context, one condition required for $M$ to have a Poisson distribution.
Assume that $M$ and $F$ can be modelled by independent Poisson distributions.
Find the probability that the total number of students who leave to study engineering in a particular year is more than 3 .
Given that the total number of students who leave to study engineering in a particular year is more than 3 , find the probability that no female students leave to study engineering in that year.

CAIE S2 2012 June Q5

10 marks Standard +0.3

5 A random variable $X$ has the distribution $\operatorname { Po } ( 3.2 )$.

A random value of $X$ is found.
1. Find $\mathrm { P } ( X \geqslant 3 )$.
2. Find the probability that $X = 3$ given that $X \geqslant 3$.
3. Random samples of 120 values of $X$ are taken.
  (a) Describe fully the distribution of the sample mean.
  (b) Find the probability that the mean of a random sample of size 120 is less than 3.3.

CAIE S2 2004 June Q6

8 marks Standard +0.8

6 At a certain airfield planes land at random times at a constant average rate of one every 10 minutes.

Find the probability that exactly 5 planes will land in a period of one hour.
Find the probability that at least 2 planes will land in a period of 16 minutes.
Given that 5 planes landed in an hour, calculate the conditional probability that 1 plane landed in the first half hour and 4 in the second half hour.

CAIE S2 2013 June Q4

6 marks Standard +0.8

4 The independent random variables $X$ and $Y$ have the distributions $\operatorname { Po } ( 2 )$ and $\operatorname { Po } ( 3 )$ respectively.

Given that $X + Y = 5$, find the probability that $X = 1$ and $Y = 4$.
Given that $\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )$, show that $3 \times 2 ^ { r - 1 } = r$ ! and verify that $r = 4$ satisfies this equation.

Edexcel S2 2022 June Q5

10 marks Challenging +1.2

The number of particles per millilitre in a solution is modelled by a Poisson distribution with mean 0.15

A randomly selected 50 millilitre sample of the solution is taken.

Find the probability that
1. exactly 10 particles are found,
2. between 6 and 11 particles (inclusive) are found. Petra takes 12 independent samples of $m$ millilitres of the solution.
  The probability that at least 2 of these samples contain no particles is 0.1184
Using the Statistical Tables provided, find the value of $m$

Pre-U Pre-U 9795/2 2012 June Q4

10 marks Challenging +1.3

4

The random variable $X$ has the distribution $\operatorname { Po } ( \lambda )$. Prove that the probability generating function, $\mathrm { G } _ { X } ( t )$, is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
The independent random variables $X$ and $Y$ have distributions $\operatorname { Po } ( \lambda )$ and $\operatorname { Po } ( \mu )$ respectively. Use probability generating functions to show that the distribution of $X + Y$ is $\operatorname { Po } ( \lambda + \mu )$.
Given that $X \sim \operatorname { Po } ( 1.5 )$ and $Y \sim \operatorname { Po } ( 2.5 )$, find $\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )$.

Pre-U Pre-U 9795/2 2013 June Q3

9 marks Standard +0.3

3

Given that $X \sim \operatorname { Po } ( 5 )$, find $\mathrm { P } ( X > 6 \mid X > 3 )$.
Given that $Y \sim \operatorname { Po } ( \lambda )$ and $\mathrm { P } ( Y \leqslant 1 ) = \frac { 1 } { 2 }$, show that $\lambda$ satisfies the equation $\lambda = \ln \{ 2 ( 1 + \lambda ) \}$.
Starting with a suitable approximation from the table of cumulative Poisson probabilities, use iteration to find $\lambda$ correct to 3 decimal places.

Pre-U Pre-U 9795/2 2016 Specimen Q4

10 marks Challenging +1.2

4

The random variable $X$ has the distribution $\operatorname { Po } ( \lambda )$. Prove that the probability generating function, $\mathrm { G } _ { X } ( t )$, is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }$$
The independent random variables $X$ and $Y$ have distributions $\operatorname { Po } ( \lambda )$ and $\operatorname { Po } ( \mu )$ respectively. Use probability generating functions to show that the distribution of $X + Y$ is $\operatorname { Po } ( \lambda + \mu )$.
Given that $X \sim \operatorname { Po } ( 1.5 )$ and $Y \sim \operatorname { Po } ( 2.5 )$, find $\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )$.

Pre-U Pre-U 9795/2 Specimen Q10

4 marks Standard +0.3

10

$X , Y$ and $Z$ are independent random variables having Poisson distributions with means $\lambda , \mu$ and $\lambda + \mu$ respectively. Find $\mathrm { P } ( X = 0$ and $Y = 2 ) , \mathrm { P } ( X = 1$ and $Y = 1 )$ and $\mathrm { P } ( X = 2$ and $Y = 0 )$. Hence verify that $\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )$.
In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
1. during a particular month both the male absence rate and the female absence rate are equal to 3,
2. during a particular month the total of the male and female absence rates is equal to 6,
3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .

Pre-U Pre-U 9795/2 Specimen Q9

10 marks Standard +0.3

A certain type of fossil occurs at a mean rate of $0.5$ per square metre at a particular location.

State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
Find the probability of at least 7 of these fossils occurring in an area of $10 \text{ m}^2$. [2]
Given that at least 4 such fossils have occurred in an area of $5 \text{ m}^2$, find the probability that there will be more than 6 found in this area of $5 \text{ m}^2$. [3]
Find the least area that must be searched in order that the probability of finding at least one fossil of this type is greater than $0.999$. Give your answer to the nearest square metre. [4]