At a funfair, Amy pays $1 for two attempts to make a bell ring by shooting at it with a water pistol. • If she makes the bell ring on her first attempt, she receives $3 and stops playing. This means that overall she has gained $2. • If she makes the bell ring on her second attempt, she receives $1.50 and stops playing. This means that overall she has gained $0.50. • If she does not make the bell ring in the two attempts, she has lost her original $1. The probability that Amy makes the bell ring on any attempt is 0.2, independently of other attempts.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [2] P(loses $1) = P(F and F) = 0.8 × 0.8. 0.8 x 0.8 or (1 – 0.2)(1-0.2) or P(F) × P(F) or P(F)+P(F) seen or implied M1 = 0.64 AG. Must see probabilities multiplied together with final answer and a clear probability statement or implied by labelled tree diagram A1
2. [4] Amount gained () -1 linked with 0.64 in table B1 Amount gained () 0.50. 0.5 seen in table B1 Prob 0.16. 0.16 seen in table linked to their 0.5 B1 Amount gained ($) 2 Prob 0.2. FT P(2.00 gained) = 0.36 – P(0.50 gained) or correct, and all amount gained linked correctly in table B1
3. [1] E(winnings) = −1 × 0.64 + 0.5 × 0.16 + 2 × 0.2 = -()0.16, -16 cents. FT Accept ()0.16 or 16 cents loss. FT unsimplified E(winnings) from their table provided Σp = 1 B1