20N.2.AHL.TZ0.H_5

pestleMathematicsAIHLPaper 220N· ahl-4-17-poisson-distributionsource ↗

A survey of British holidaymakers found that $15 %$ of those surveyed took a holiday in the Lake District in $2019$ .

A random sample of $16$ British holidaymakers was taken. The number of people in the sample who took a holiday in the Lake District in $2019$ can be modelled by a binomial distribution.

State two assumptions made in order for this model to be valid.

[2]

a.i.

Find the probability that at least three people from the sample took a holiday in the Lake District in $2019$ .

[2]

a.ii.

From a random sample of $n$ holidaymakers, the probability that at least one of them took a holiday in the Lake District in $2019$ is greater than $0.999$ .

Determine the least possible value of $n$ .

[3]

Markscheme / solution

people’s holidays are independent of each other R1

the proportion is constant (at $0.15$ ) R1

[2 marks]

a.i.

$Χ ~ B (16, 0.15)$

$P (Χ \geq 3) = 0.439$ (M1)A1

[2 marks]

a.ii.

probability of at least one $= 1 -$ probability of none

$\Rightarrow 1 - 0 . 85^{n} > 0.999$ OR $0 . 85^{n} < 0.001$ (A1)

attempt to solve inequality (M1)

$n \geq 42.503 \dots$

so least possible $n = 43$ A1

[3 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]