A-LevelMathematicsProbability and statistics9709/53 · May/June 2021 · Q711 Marks

In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800. Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table. [Table: Quality of broadband service (Excellent, Good, Poor) for Reeta, Shan, Teber] [Table data: Reeta: Excellent=75, Good=118, Poor=32; Shan: Excellent=223, Good=177, Poor=40; Teber: Excellent=12, Good=60, Poor=63]

(a)(i)Find the probability that a randomly chosen household is in Shan and has poor broadband service.[1]
(a)(ii)Find the probability that a randomly chosen household has good broadband service given that the household is in Shan.[2]
(b)(i)In the whole of Arka there are a large number of households. A survey showed that 35% of households in Arka have no broadband service. 10 households in Arka are chosen at random. Find the probability that fewer than 3 of these households have no broadband service.[3]
(b)(ii)120 households in Arka are chosen at random. Use an approximation to find the probability that more than 32 of these households have no broadband service.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[1] B1 40/800 or 1/20 or 0.05
[2] M1 Their 223 + 177 + 40 seen as denominator of fraction in the final answer, accept unsimplified
A1 CAO
[3] M1 One term: ¹⁰Cxpˣ(1 – p)¹⁰⁻ˣ for 0 < x < 10, any 0<p<1
A1 Correct unsimplified expression, or better
A1 0.262
[5] B1 Correct mean and variance seen, allow unsimplified
M1 Substituting their mean and variance into standardisation formula (any number), condone σ² or √σ
M1 Using continuity correction 31.5 or 32.5
M1 Appropriate area Φ, from final process, must be probability
A1 0.965 <p ≤ 0.966

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