$P(\text{win on a throw})=\frac{2}{6}=\frac{1}{3}$, $P(\text{lose})=\frac{2}{3}$.

The player stops as soon as he wins or after 3 throws.

• Win on throw 1: prob $=\frac{1}{3}$; gain $=+100$.
• Win on throw 2 (lost throw 1): prob $=\frac{2}{3}\cdot\frac{1}{3}=\frac{2}{9}$; net $=-50+100=+50$.
• Win on throw 3 (lost throws 1,2): prob $=\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{1}{3}=\frac{4}{27}$; net $=-100+100=0$.
• Lose all 3 throws: prob $=\left(\frac{2}{3}\right)^3=\frac{8}{27}$; net $=-150$.

$E = \frac{1}{3}(100)+\frac{2}{9}(50)+\frac{4}{27}(0)+\frac{8}{27}(-150)$

$=\frac{100}{3}+\frac{100}{9}+0-\frac{1200}{27}$

$=\frac{900}{27}+\frac{300}{27}-\frac{1200}{27}=\frac{0}{27}=\mathbf{0}$