Question 6 - A-Level Maths

CAIE FP2 2012 June — Question 6 6 marks

Exam Board	CAIE
Module	FP2 (Further Pure Mathematics 2)
Year	2012
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Geometric Distribution
Type	Find minimum n for P(X ≤ n) > threshold
Difficulty	Moderate -0.8 This is a straightforward application of the geometric distribution with standard formulas. Part (i) requires direct substitution into P(X=k)=(1-p)^(k-1)×p, part (ii) is immediate recall that E(N)=1/p=100, and part (iii) uses the cumulative distribution formula 1-(1-p)^n>0.9, requiring only logarithms to solve. All three parts are routine textbook exercises with no problem-solving insight required, making this easier than average despite being Further Maths content.
Spec	5.02f Geometric distribution: conditions 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2

The probability that a particular type of light bulb is defective is \(0.01\). A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. [2] The first defective bulb is the \(N\)th to be tested. Write down the value of E(\(N\)). [1] Find the least value of \(n\) such that P(\(N \leqslant n\)) is greater than \(0.9\). [3]

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Question 6:

Answer	Marks
6	th

Find prob. that 10 bulb is first defective

one: (1 – 0⋅01) 9 × 0⋅01 = 0⋅00914

(allow 0⋅00913) M1 A1

State or find E(N): E(N) = 1/0⋅01 = 100 B1

Formulate condition for n: P(N ≤ n) = 1 – P(N > n)

(equality throughout loses this M1 only) = 1 – 0⋅99 n > 0⋅9, 0⋅99 n < 0⋅1 M1

Take logs (any base) to give inequality for n: n > log 0⋅1 / log 0⋅99 M1

Answer	Marks
Find nmin: n > 229⋅1, nmin = 230 A1	2

1

Answer	Marks	Guidance
3	[6]
Page 6	Mark Scheme: Teachers’ version	Syllabus
GCE A LEVEL – May/June 2012	9231	22

Question 6:
6 | th
Find prob. that 10 bulb is first defective
one: (1 – 0⋅01) 9 × 0⋅01 = 0⋅00914
(allow 0⋅00913) M1 A1
State or find E(N): E(N) = 1/0⋅01 = 100 B1
Formulate condition for n: P(N ≤ n) = 1 – P(N > n)
(equality throughout loses this M1 only) = 1 – 0⋅99 n > 0⋅9, 0⋅99 n < 0⋅1 M1
Take logs (any base) to give inequality for n: n > log 0⋅1 / log 0⋅99 M1
Find nmin: n > 229⋅1, nmin = 230 A1 | 2
1
3 | [6]
Page 6 | Mark Scheme: Teachers’ version | Syllabus | Paper
GCE A LEVEL – May/June 2012 | 9231 | 22

Show LaTeX source

The probability that a particular type of light bulb is defective is $0.01$. A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective.
[2]

The first defective bulb is the $N$th to be tested. Write down the value of E($N$).
[1]

Find the least value of $n$ such that P($N \leqslant n$) is greater than $0.9$.
[3]

\hfill \mbox{\textit{CAIE FP2 2012 Q6 [6]}}

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