A-LevelMathematicsProbability and statistics9709/52 · Feb/Mar 2021 · Q13 Marks

A fair spinner with 5 sides numbered 1, 2, 3, 4, 5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.

(a)Find the probability that a score of 3 is obtained for the first time on the 8th spin.[1]
(b)Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time.[2]

Mark scheme

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

[1] B1 Evaluated, final answer.
[1] M1 1-p^n = 5,6 or p + pq + pq²+pq³+pq^ (+ pq⁵) 0<p<1, p + q = 1, Sum of a geometric series may be used.
[1] A1 Final answer.
[1] M1 (p)⁵ + ⁵C₄(p)⁴(q)¹ + ⁵C₃(p)³(q)² + ⁵C₂(p)²(q)³ + ⁵C₁(p)¹(q)⁴ or (p)⁶ + ⁶C₅(p)⁵(q)¹ + ⁶C₄(p)⁴(q)² + ⁶C₃(p)³(q)³ +⁶C₂(p)²(q)⁴ + ⁶C₁(p)¹(q)⁵, 0 < p < 1, p + q = 1 At least first, last and one intermediate term is required to show pattern of terms if not all terms stated.
[1] A1 Final answer.

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 3 marks.
Source: A-Level Mathematics, Feb/Mar 2021 Paper 5 Variant 2, question 1.

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