Birgitte has a six-sided dice. She suspects that the dice is biased so that the probability, p, that it will show a six on one throw is less than 1/6. She throws the dice 30 times and finds that it shows a six on exactly 2 throws. Later, Birgitte carries out a similar test at the 5% significance level, using another 30 throws of the dice.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] Ho: p = 1/6 H₁: p < 1/6
2. [M1] Expression or terms must be seen. No end errors.
3. [A1] = 0.103 SC Unsupported correct answer scores B1.
4. [M1] '0.103' > 0.05 Valid comparison (must be a tail comparison and from a Bin but not necessarily correct Bin).
5. [A1FT] [Accept Ho] Insufficient evidence to suggest that the probability is less than 1/6 Or Insufficient evidence to support Birgitte's suspicion FT their 0.103. No contradictions, in context and non-definite. Note: Condone 'Insufficient evidence to suggest that the dice is biased' scores A1.
6. [M1] P(X < 1) attempted using B(30, 1/6). No end errors.
7. [A1] P(0) and P(1) expression or terms may be seen in (a). If not in (a) unsupported answer of 0.0295 scores B1.
8. [M1] Use of B(30,0.02) to find 1– P(X < 1). May be implied.
9. [M1] = 1- (0.98^30 + 30 * 0.98^29 * 0.02) = 1 – (0.54548 + 0.33397) No end errors. Expression or terms must be seen.
10. [A1] = 0.121 (3 sf) SC Unsupported correct answer scores M1B1.