Boxing Tournament

Since there are 5 matches, we consider all the result of his first match until the fifth match. He wins at most twice if after 5 matches he has zero win or one win or two wins.

p = probability of winning = $\frac{2}{3}$

q = probability of losing = 1 - $\frac{2}{3}$ = $\frac{1}{3}$

n = number of matches

r = number of exact wins

$P$ = $(\frac{1}{3})^5$ + $5C1(\frac{2}{3})^1(\frac{1}{3})^4$ + $5C2(\frac{2}{3})^2(\frac{1}{3})^3$ = $\frac{1}{243} + \frac{10}{243} + \frac{51}{243}$ = $\frac{17}{81}$