A2 June 2019 Q1

EdexcelCurrent spec6 marksGeometric & Negative Binomial

1. A chocolate manufacturer places special tokens in 2% of the bars it produces so that each bar contains at most one token. Anyone who collects 3 of these tokens can claim a prize.

Andreia buys a box of 40 bars of the chocolate.

(a) Find the probability that Andreia can claim a prize. (2)

Barney intends to buy bars of the chocolate, one at a time, until he can claim a prize.

(b) Find the probability that Barney can claim a prize when he buys his 40th bar of chocolate. (3)

Mark scheme (a)

Scheme	Marks	AO
[Let \(X\) = no. of prizes Andreia wins] \(X \sim \mathrm{B}(40, 0.02)\)	M1	3.3
[Require \(\mathrm{P}(X \geqslant 3) = 1 - \mathrm{P}(X \leqslant 2)\)] \(= 0.04567\ldots\) awrt 0.0457	A1	1.1b
	(2)

Notes

M1 for selecting a suitable model i.e. \(\mathrm{B}(40, p)\) where \(p\) is any probability
Written or used, may be implied by a correct ans or \(0.037429\ldots\) from \(\mathrm{P}(X = 3)\)

A1 for awrt 0.0457 (correct answer only 2/2)

Mark scheme (b)

Scheme	Marks	AO
[Let \(Y\) = no. of the bar when Barney wins] \(Y \sim \mathrm{NegBin}(3, 0.02)\)	M1	3.3
\([\mathrm{P}(Y = 40) =]\ \dbinom{39}{2} \times 0.02^2 \times 0.98^{37} \times 0.02\)	M1	3.4
\(= 0.0028071\ldots\) awrt 0.00281	A1	1.1b
	(3)

Notes

1^st M1 for selecting a suitable model (\(\mathrm{NB}(3, 0.02)\)) May be implied by a correct expression

2^nd M1 for use of model to form a correct expression

SC \(p \ne 0.02\) Allow prob of the form \(\dbinom{39}{2} p^3 (1 - p)^{37}\) where \(0 \lt p \lt 1\) scores M0M1

A1 for awrt 0.00281 (accept awrt \(2.81 \times 10^{-3}\)) [correct answer with no working scores 3/3]

Scheme	Marks	AO
\(\mathrm{E}(Y) = \dfrac{3}{0.02} = \underline{\mathbf{150}}\)	B1	1.1b
	(1)
	(6 marks)