In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable X is the number of darts in a turn that hit the bull's-eye. The probability distribution of X is given in the following table. X | 0 | 1 | 2 | 3 P(X = x) | 0.6 | p | q | 0.05 It is given that E(X) = 0.55.

Mark scheme

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

1. [B1] Sum of probabilities = 1.
2. [B1] Use given information.
3. [M1] Either a single expression with one variable eliminated formed or two expressions with both variables on the same side seen with at least one variable value stated.
4. [A1] CAO, both WWW
5. [SC] If M0 with correct answers SC B1.
6. [M1] Appropriate variance formula including (E(X))², accept unsimplified.
7. [A1] CAO (must be exact).
8. [M1] One correct term: ¹²Cx pˣ (1 – p)¹²⁻ˣ for 0 < x < 12, 0 < p < 1.
9. [A1FT] Correct unsimplified expression, or better in final answer. Unsimplified expression must be seen to FT their p from 6(a) or correct.
10. [A1]
11. [B1] Evaluated.