# Probability with Wins, Draws and Losses in Go Matches

In a Go match between the Amy team and Ben team, a game is played on each of board $1$, board $2$, board $3$ and board $4$.

On each board, the probability that the Amy team wins is $0.2$, the probability of a draw is $0.6$ and the probability that the Ben team loses is $0.2$.

The results are recorded by listing the outcomes of the games for the Amy team in board order. For example, if the Amy team wins on board $1$, draws on board $2$, loses on board $3$ and wins on board $4$, the result is recorded as $WDLW$.

(a)   How many different recordings are possible?

$3 \times 3 \times 3 \times 3 = 3^4$

(b)   Calculate the probability of the result which is recorded as $WDDL$.

$0.2 \times 0.6 \times 0.6 \times 0.2 = 0.0144$

Teams score $2$ points for each game won, $1$ a point for each game drawn and $0$ points for each game lost. Find the probability that the Amy team scores more points than the Ben team.

(c)   Find the probability that the Amy team scores more points than the Ben team.

$\displaystyle \begin{array}{|c|l|} \hline WWWW & 0.2^4 \\ \hline WWWD & ^4C_3 \times 0.2^3 \times 0.6 \\ \hline WWWL & ^4C_3 \times 0.2^3 \times 0.2 \\ \hline WWDL & ^4C_2 \times 0.2^2 \times 0.6 \times 0.2 \\ \hline WWDD & \dfrac{4!}{2!2!} \times 0.2^2 \times 0.6^2 \\ \hline WDDD & ^4C_1 \times 0.2 \times 0.6^3 \\ \hline \end{array}$
The sum of these probabilities is $0.3152$. 