Question 15 - A-Level Maths

AQA Paper 3 2018 June — Question 15 7 marks

Exam Board	AQA
Module	Paper 3 (Paper 3)
Year	2018
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Binomial Distribution
Type	E(X) and Var(X) with probability calculations
Difficulty	Easy -1.3 This is a straightforward binomial distribution question requiring only standard recall and basic calculations. Parts (a) and (d) are direct recall, part (b) is a single probability calculation, part (c) uses complement rule (routine technique), and part (e) asks for standard binomial assumptions. No problem-solving or novel insight required—purely textbook application.
Spec	2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities

Abu visits his local hardware store to buy six light bulbs. He knows that 15% of all bulbs at this store are faulty.

State a distribution which can be used to model the number of faulty bulbs he buys. [1 mark]
Find the probability that all of the bulbs he buys are faulty. [1 mark]
Find the probability that at least two of the bulbs he buys are faulty. [2 marks]
Find the mean of the distribution stated in part (a). [1 mark]
State two necessary assumptions in context so that the distribution stated in part (a) is valid. [2 marks]

Show mark scheme Show mark scheme source

Question 15:

Answer	Marks	Guidance
15(a)	States the correct binomial
distribution	AO3.3	B1

Answer	Marks	Guidance
15(b)	Calculates the correct probability	AO1.1b

Answer	Marks	Guidance
15(c)	Calculates or
using the binomial distribution	AO1.1a	M1

𝑃𝑃(𝑋𝑋 ≥ 2)= 1−𝑃𝑃(𝑋𝑋 ≤ 1)

( )

𝑃𝑃(𝑋𝑋 ≤ 1) 𝑃𝑃(𝑋𝑋 ≤ 2)

Answer	Marks	Guidance
Obtains the correct answer	AO1.1b	A1

Answer	Marks	Guidance
15(d)	Finds the correct mean	AO1.1b

0.9

Answer	Marks	Guidance
15(e)	States a first appropriate
assumption in context	AO3.5b	B1

faulty is fixed

A light bulb being faulty is

independent of any other light bulb

being faulty

States a second appropriate

Answer	Marks	Guidance
assumption in context	AO3.5b	B1
Total	7
Q	Marking Instructions	AO

Question 15:
--- 15(a) ---
15(a) | States the correct binomial
distribution | AO3.3 | B1 | B(6, 0.15)
--- 15(b) ---
15(b) | Calculates the correct probability | AO1.1b | B1 | 0.0000114
--- 15(c) ---
15(c) | Calculates or
using the binomial distribution | AO1.1a | M1 | 𝑃𝑃(𝑋𝑋 ≤ 1) = 0.7764
𝑃𝑃(𝑋𝑋 ≥ 2)= 1−𝑃𝑃(𝑋𝑋 ≤ 1)
( )
𝑃𝑃(𝑋𝑋 ≤ 1) 𝑃𝑃(𝑋𝑋 ≤ 2)
Obtains the correct answer | AO1.1b | A1
--- 15(d) ---
15(d) | Finds the correct mean | AO1.1b | B1 | 𝑃𝑃 𝑋𝑋 ≥ 2 = 0.224
0.9
--- 15(e) ---
15(e) | States a first appropriate
assumption in context | AO3.5b | B1 | The probability of a light bulb being
faulty is fixed
A light bulb being faulty is
independent of any other light bulb
being faulty
States a second appropriate
assumption in context | AO3.5b | B1
Total | 7
Q | Marking Instructions | AO | Marks | Typical Solution

Show LaTeX source

Abu visits his local hardware store to buy six light bulbs.

He knows that 15% of all bulbs at this store are faulty.

\begin{enumerate}[label=(\alph*)]
\item State a distribution which can be used to model the number of faulty bulbs he buys.
[1 mark]

\item Find the probability that all of the bulbs he buys are faulty.
[1 mark]

\item Find the probability that at least two of the bulbs he buys are faulty.
[2 marks]

\item Find the mean of the distribution stated in part (a).
[1 mark]

\item State two necessary assumptions in context so that the distribution stated in part (a) is valid.
[2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2018 Q15 [7]}}

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Q1 1 Q2 1 Q3 1 Q4 3 Q5 3 Q6 13 Q7 5 Q8 9 Q9 7 Q10 10 Q11 1 Q12 1 Q13 3 Q14 6 Q15 7 Q16 12 Q17 12 Q18 8