21N.2.SL.TZ0.4

pestleMathematicsAASLPaper 221N· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

A factory manufactures lamps. It is known that the probability that a lamp is found to be defective is $0.05$ . A random sample of $30$ lamps is tested.

Find the probability that there is at least one defective lamp in the sample.

[3]

Given that there is at least one defective lamp in the sample, find the probability that there are at most two defective lamps.

[4]

Markscheme / solution

recognize that the variable has a Binomial distribution (M1)

$X ~ B (30, 0.05)$

attempt to find $P (X \geq 1)$ (M1)

$1 - P (X = 0)$ OR $1 - 0 . 95^{30}$ OR $1 - 0.214638 \dots$ OR $0.785361 \dots$

Note: The two M marks are independent of each other.

$P (X \geq 1) =0.785$ A1

[3 marks]

recognition of conditional probability (M1)

$P (X \leq 2 X \geq 1)$ OR $P (at most 2 defective | at least 1 defective)$

Note: Recognition must be shown in context either in words or symbols but not just $P (A B)$ .

$\frac{P (1 \leq X \leq 2)}{P (X \geq 1)}$ OR $\frac{P (X = 1) + P (X = 2)}{P (X \geq 1)}$ (A1)

$\frac{0.597540 \dots}{0.785361 \dots}$ OR $\frac{0.812178 \dots - 0.214638 \dots}{0.785361 \dots}$ OR $\frac{0.338903 \dots + 0.258636 \dots}{0.785361 \dots}$ (A1)

$= 0.760847 \dots$

$P (X \leq 2 X \geq 1) = 0.761$ A1

[4 marks]

Examiners’ report

[N/A]