Question

Football teams $T_1$ and $T_2$ have to play two game against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2$ respectively, after two games.
$P(X>Y)$ is

• A. $\frac{1}{4}$
• B. $\frac{5}{12}$
• C. $\frac{1}{2}$
• D. $\frac{7}{27}$

Answer 1

Answer: (B)

$P(X>Y)=P(WW)+P(WD)+P(DW)$
$=\frac{1}{2}\times \frac{1}{2}+\frac{1}{2}\times \frac{1}{6}+\frac{1}{6}\times \frac{1}{2}=\frac{5}{12}$