For a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday.

Mark scheme

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

1. [B1] 0.03457...
2. [M1] One term 10Cx (p)^x (1-p)^(10-x). With 0 < p < 1, x ≠ 0 or 10.
3. [A1] Correct un-simplified expression. Allow 10 for 10C1.
4. [B1] 0.838 < p < 0.839.
5. [M1] One term 10Cx (p)^x (1-p)^(10-x). With 0 < p < 1, x ≠ 0 or 10.
6. [A1] Correct un-simplified expression. Allow 10 for 10C1. Condone omission of up to 5 of the middle 6 terms. Condone omission of last bracket only. If both brackets omitted in un-simplified expression allow recovery for final stated calculation of 1 – 0.1616 or final answer WRT to 0.8384.
7. [B1] 0.838 < p < 0.8385.
8. [B1] 56 and 48 seen, allow un-simplified, may be seen in the standardisation formula. (σ = √48, 4√3, 6.928 ≤ σ ≤ 6.9283 implies correct variance.
9. [M1] Substituting their μ and positive σ into the ± standardising formula (any number for 65.5), allow σ^2 or √σ.
10. [M1] Use continuity correction 64.5 or 65.5 in standardisation formula. Note: If no standardisation formula seen ±9.5/√48 or ±9.5/6.928 scores M2.
11. [M1] Appropriate area Φ, from final process. Must be a probability. May be implied by a sketch of the required probability area. Note: correct final answer implies this M1. Expect final answer < 0.5.
12. [A1] Final answer. Accept 0.08505 < p < 0.0852.