jmc

For the contest to go to 7 games, the teams must be tied 3-3 after 6 games. There are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ ways to pick which 3 of the 6 games the Lakers will win. Then there is a ${\left(\frac{1}{3}\right)}^3{\left(\frac{2}{3}\right)}^3$ chance that they will win the 3 games we pick and lose the other 3. So there is a $20{\left(\frac{1}{3}\right)}^3{\left(\frac{2}{3}\right)}^3=\frac{160}{729}$ chance that the contest becomes tied 3-3. Then, there is a $\frac{1}{3}$ chance that the Lakers win the last game. So the final probability is $\frac{160}{729}\cdot \frac{1}{3}=\boxed{\frac{160}{2187}}$.