To find $P(X=5)$, we need to consider the following cases where the maximum score is 5:
1. One of the scores is 5 , and the other is 4 .
2. Both scores are 5 .
\section*{1. Both tests result in a score of 5 :}
$$
P(X 1=5 \text { and } X 2=5)=\left(\frac{1}{3}\right) \times\left(\frac{1}{3}\right)=\frac{1}{9}
$$
\section*{2. One test results in 5 and the other in 4:}
$$
\begin{aligned}
& P(X 1=5 \text { and } X 2=4)=\left(\frac{1}{3}\right) \times\left(\frac{1}{3}\right)=\frac{1}{9} \\
& P(X 1=4 \text { and } X 2=5)=\left(\frac{1}{3}\right) \times\left(\frac{1}{3}\right)=\frac{1}{9}
\end{aligned}
$$
The total probability for one score being 5 and the other being 4 is:
$\frac{1}{9}+\frac{1}{9}=\frac{2}{9}$
So, the total probability that the maximum score is 5 :
$P(X=5)=\frac{1}{9}+\frac{2}{9}=\frac{3}{9}=\frac{1}{3}$
Therefore, the probability that your final score $X$ is 5 is:
$P(X=5)=\frac{1}{3}$
