Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is 1/4, independently of all other throws. Eric throws all three coins at the same time. Events A and B are defined as follows. A: all three coins show the same result B: at least one of the biased coins shows a head

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. Method 1: Scenarios identified ignoring unbiased coin
2. [M1] All 3 different calculations seen unsimplified. P(BH1 BT2) = 1/4 × 3/4 = 3/16 P(BT1 BH2) = 3/4 × 1/4 = 3/16 P(BH1 BH2) = 1/4 × 1/4 = 1/16
3. [A1] Clear identification of all scenarios, linked probabilities and sum. AG 3/16 + 3/16 + 1/16 = 7/16
4. Method 2: Scenarios identified with all 3 coins
5. [M1] All 6 different calculations seen unsimplified. P(H BH1 BT2) = 1/2 × 1/4 × 3/4 = 3/32 P(T BH1 BT2) = 1/2 × 1/4 × 3/4 = 3/32 P(H BT1 BH2) = 1/2 × 3/4 × 1/4 = 3/32 P(T BT1 BH2) = 1/2 × 3/4 × 1/4 = 3/32 P(H BH1 BH2) = 1/2 × 1/4 × 1/4 = 1/32 P(T BH1 BH2) = 1/2 × 1/4 × 1/4 = 1/32
6. [A1] Clear identification of all scenarios, linked probabilities and sum. AG (1+3+3+1+3+3) / 32 = 14/32 = 7/16
7. Method 3: 1- P(BT1 BT2) ignoring unbiased coin
8. [M1] Calculation seen unsimplified and 1 - probability seen. 1 - (3/4)²
9. [A1] Clear identification of scenario used, linked probability and calculation. AG 1 - 9/16 = 7/16
10. Method 4: 1- P(BT1 BT2) with all 3 coins
11. [M1] Both calculations seen unsimplified and 1 - 2 probabilities seen. 1 - P(H BT, BT2) - P(T BT, BT2) = 1 - (1/2 * 3/4 * 3/4) - (1/2 * 3/4 * 3/4)
12. [A1] Clear identification of all scenarios used, linked probabilities and calculation. AG 1 - 9/32 - 9/32 = 1 - 18/32 = 14/32 = 7/16
13. [M1] Their identified P(HHH) or correct as numerator and their identified P(B) or correct as denominator. Either numerical expression acceptable.
14. [A1] Accept 0.071428... rounded to at least 3SF.
15. [B1] Table with correct X values and at least one probability. Condone any additional X values if probability stated as 0.
16. [B1] P(1) or P(2) correct, need not be in table, accept unsimplified.
17. [B1] 4 correct probabilities linked with correct outcomes, may not be in table. Decimals correct to at least 3 SF. SC B1 for 4 probabilities (0 < p < 1) sum to 1 ± 0.005 with P(1) and P(2) incorrect.