A-LevelMathematicsProbability and statistics9709/51 · May/June 2022 · Q68 Marks

Janice is playing a computer game. She has to complete level 1 and level 2 to finish the game. She is allowed at most two attempts at any level. • For level 1, the probability that Janice completes it at the first attempt is 0.6. If she fails at her first attempt, the probability that she completes it at the second attempt is 0.3. • If Janice completes level 1, she immediately moves on to level 2. • For level 2, the probability that Janice completes it at the first attempt is 0.4. If she fails at her first attempt, the probability that she completes it at the second attempt is 0.2.

(a)Show that the probability that Janice moves on to level 2 is 0.72.[1]
(b)Find the probability that Janice finishes the game.[3]
(c)Find the probability that Janice fails exactly one attempt, given that she finishes the game.[4]

Mark scheme

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

[B1] Clear identified calculation AG
[M1] 0.72 × u, 0 < u < 1
[M1] v × (0.4 + 0.6 × 0.2), or v × (1 – 0.6 × 0.8) 0 < v ≤ 1 no additional terms SC B1 for 0.72 × (0.4 + 0.12) or 0.72 × (1 – 0.48)
[A1] WWW. Condone 0.374. SC B1 for 0.3744 only
[M1] Either 0.6 × 0.6 × 0.2 or 0.4 × 0.3 × 0.4 seen Condone 0.072 or 0.048 if seen in (b)
[A1] Both correct accept unsimplified expression. No additional terms
[M1] Their sum of two 3-term probabilities as numerator / their (b) or correct
[A1] 0.3205 < p < 0.321

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