A-LevelMathematicsProbability and statistics9709/52 · Feb/Mar 2024 · Q28 Marks

Sam is a member of a soccer club. She is practising scoring goals. The probability that Sam will score a goal on any attempt is 0.7, independently of all other attempts.

(a)Sam makes 10 attempts at scoring goals. Find the probability that Sam will score goals on fewer than 8 of these attempts.[3]
(b)Find the probability that Sam's first successful attempt will be before her 5th attempt.[2]
(c)Wei is a member of the same soccer club. He is also practising scoring goals. The probability that Wei will score a goal on any attempt is 0.6, independently of all other attempts. Wei is going to keep making attempts until he scores 3 goals. Find the probability that he scores his third goal on his 7th attempt.[3]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

[M1] One term 10Cx (p)^x (1-p)^10-x with 0<p<1, x≠0 or 10.
[A1] Correct unsimplified expression. Condone omission of last bracket only.
[B1] 0.617 < p < 0.6175 www.
[M1] 1-b^d; b = 0.3, 0.7; d = 4, 5. Or 1 – c^e – (1 – c) × c^(e-1); c = 0.3, 0.7; e = 5, 6.
[A1] Condone 0.992. If M0 scored, SC B1 for 0.9919 or 9919/10000 only.
[M1] (0.4)^a (0.6)^b; r = 2, 3. No inappropriate addition.
[M1] (0.4)^a (0.6)^b × 6C2; a + b = 6, 7.
[A1] Accept 0.082944 correct to at least three significant figures. If A0 scored, SC B1 for correct answer www.

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