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} $