Anil is taking part in a tournament. In each game in this tournament, players are awarded 2 points for a win, 1 point for a draw and 0 points for a loss. For each of Anil's games, the probabilities that he will win, draw or lose are 0.5, 0.3 and 0.2 respectively. The results of the games are all independent of each other. The random variable X is the total number of points that Anil scores in his first 3 games in the tournament.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [M1] 0.5×0.2^2 ×3C(a) (or 3)+x; 0 < x < 1; a = 1, 2. Or 0.3^2 ×0.2×3C(b) (or 3) + y; 0 < y < 1; b = 1, 2. Or 0.5×0.2^2 × a +0.3^2 ×0.2×b; a, b = 1, 2, 3.
2. [A1] AG. Fully correct solutions with outcomes identified and linked to appropriate probabilities. Condone 2 = W, 1 = D, 0 = L. Probabilities alone do not identify outcomes. If individual scenarios are identified, separate calculations must correspond to the order.
3. [B1] One additional correct probability in table or clearly identified.
4. [B1] A second additional correct probability in table or clearly identified.
5. [B1] Final correct probability, all probabilities in table. If 0/3 scored, SC B1 for three additional probabilities in table that sum to 0.269 exactly.
6. [M1] Accept unsimplified expression. May be calculated in the variance, FT their table with probabilities, 0 < p < 1, that sum to 1. FT acceptable at the bold partially evaluated stage.
7. [M1] Appropriate variance formula using their (E(X))^2 value. FT their table with probabilities, 0<p<1, that may not sum to 1.
8. [A1] Cao. Condone 183/100.