There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table. | | Swimming | Cycling | Running | |-----------|----------|---------|---------| | Female | 104 | 50 | 66 | | Male | 31 | 57 | 92 | A student is chosen at random. On average at all the schools in this country 30% of the students do not like any sports.

Mark scheme

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

1. [1] B1 Evaluated, exact value.
2. [1] M1 Their P(M) × their P(S) seen, accept unsimplified.
3. [1] A1 P(M), P(S) and P(M∩ S) notation seen, numerical comparison and correct conclusion, WWW.
4. [1] M1 [P(M|S) =] their P(M∩S) / their P(S) (oe) seen, accept unsimplified.
5. [1] A1 P(M), P(S) and P(M∩ S) notation seen, numerical comparison and correct conclusion, WWW.
6. [1] M1 ¹⁰Cₓpˣ(1-p)¹⁰⁻ˣ for 0 < x < 10, 0 < p < 1, any p.
7. [1] A1 Correct expression, accept unsimplified, condone omission of final bracket, condone recovery from poor notation.
8. [1] A1 Accept 0.61715 ≤ p < 0·61722, WWW.
9. [1] M1 ¹⁰Cₓpˣ(1-p)¹⁰⁻ˣ for 0 < x < 10, 0 < p < 1, any p.
10. [1] A1 Correct unsimplified expression.
11. [1] A1 Accept 0.61715 < p < 0·61722, WWW.
12. [1] B1 Correct mean and variance, allow unsimplified. Condone σ = 4·347 evaluated.
13. [1] M1 Substituting their μ and σ (not σ², √σ) into the ±standardising formula with a numerical value for '31·5'.
14. [1] M1 Using either 31.5 or 32.5 within a standardising formula with numerical values for their μ and σ (condone σ², √σ).
15. [1] M1 Appropriate area Φ, from standardisation formula P(z<...) in final solution, must be probability.
16. [1] A1 Allow 0·8495 < p < 0·85(0), final answer WWW.