Jacob throws three coins at the same time. The first coin is biased so that the probability of obtaining a head when it is thrown is 1/3. The second coin is biased so that the probability of obtaining a head when it is thrown is 1/4. The third coin is biased so that the probability of obtaining a head when it is thrown is 1/5. The random variable X is the number of heads obtained.

Mark scheme

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

1. [1] 1/3 × 1/4 × 4/5 + 1/3 × 3/4 × 1/5 + 2/3 × 1/4 × 1/5 = 9/60 = 3/20 AG
2. Order of coins must be consistent with question if not identified.
3. [1] Table with correct values of x and at least two correct non-zero probabilities.
4. [1] One more correct non-zero probability linked with correct x value, need not be in table if clearly identified, accept unsimplified (total of 3 correct probabilities).
5. [1] 4 correct probabilities linked with the correct outcomes, may not be in table. Decimals correct to at least 3SF. SC1 for 4 or more probabilities summing to 1 placed in a probability distribution table.
6. x: 0, P(X=x): 24/60, 0.4
7. x: 1, P(X=x): 26/60, 0.433
8. x: 2, P(X=x): 9/60, 0.15
9. x: 3, P(X=x): 1/60, 0.0167
10. [1] Appropriate variance formula using (E(X))² value. FT their table with 3 or more probabilities (0 < p < 1) which need not sum to 1, with an expression no more evaluated than in bold. FT acceptable at the bold partially evaluated stage with their probabilities.
11. [1] 0.5695 < Var(X) < 0.570. If M0 scored, SC1 for 2051/3600 = 0.570 WWW Note: 0.57 without more accurate previous value penalised as 2SF.
12. Var(X) = (0² × 24 + 1² × 26 + 2² × 9 + 3² × 1) / 60 – (47/60)² = 2051/3600 ≈ 0.5697.