Question

Football teams $T_1$ and $T_2$ have to play two games 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 points 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. $\frac{11}{36}$
• B. $\frac{1}{3}$
• C. $\frac{13}{36}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C

$\frac{13}{36}$

$P(X=Y)$

$=P\left(\text{draw}\right)\cdot P\left(\text{draw}\right)+P\left(T_1\text{win}\right)\cdot P\left(T_2\text{win}\right)+P\left(T_2\text{win}\right)\cdot P\left(T_1\text{win}\right)$

$=(\frac{1}{6}\times \frac{1}{6})+(\frac{1}{2}\times \frac{1}{3})+(\frac{1}{3}\times \frac{1}{2})$

$=\frac{13}{36}$