A-LevelMathematicsProbability and statistics9709/52 · Feb/Mar 2022 · Q26 Marks

In a certain country, the probability of more than 10cm of rain on any particular day is 0.18, independently of the weather on any other day.

(a)Find the probability that in any randomly chosen 7-day period, more than 2 days have more than 10 cm of rain.[3]
(b)For 3 randomly chosen 7-day periods, find the probability that exactly two of these periods have at least one day with more than 10 cm of rain.[3]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.

M1 One term ⁷Cx pˣ (1-p)⁷⁻ˣ, 0<p<1,0<x<7
A1 Correct unsimplified expression or better Condone omission of brackets if recovered
B1 WWW. 0.115 < p < 0.1155 not from wrong working
P(X > 2) = 1 - P(X ≤ 2)
= 1 - [P(X=0) + P(X=1) + P(X=2)]
= 1 - [⁷C₀ (0.18)⁰ (0.82)⁷ + ⁷C₁ (0.18)¹ (0.82)⁶ + ⁷C₂ (0.18)² (0.82)⁵]
= 1 - [0.249285 + 0.383048 + 0.252251]
= 1 - 0.884584 = 0.115416
B1 AWRT 0.751 seen
M1 FT their 1-p⁷ or their 0.7507 if identified, not 0.18, 0.82 Accept ×3Cr, r=1,2 or ×3P1 for ×3 Condone ×2
A1 Accept 0.421 < p < 0.4215 SC B1 if 0/3 scored for final answer only 0.421 < p < 0.4215
P(at least 1 day of rain in a 7-day period) = 1 - P(0 days) = 1 - (0.82)⁷ = 1 - 0.249285 = 0.750715
P(exactly 2 periods) = ³C₂ (0.750715)² (1 - 0.750715)¹
= 3 × (0.750715)² × (0.249285) = 0.421067

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