Suppose the man fires n times and let $\mathrm{X}$ denote the number of times he hits the target. Then, $$ \mathrm{P}(\mathrm{X}=\mathrm{r})=\mathrm{nc}\left(\frac{1}{4}\right)^{\mathrm{r}}\left(\frac{3}{4}\right)^{\mathrm{r}-\mathrm{r}}, \mathrm{r}=0,1,2,-\mathrm{n} $$ If is given that $P(X, 1)\gt\frac{2}{3}$

$$ 1-\mathrm{P}(\mathrm{X}=0)\gt\frac{2}{3} $$ $$ \begin{array}{l} 1-n c\left(\frac{1}{4}\right)^{0}\left(\frac{3}{4}\right)^{n}\gt\frac{2}{3} \\ 1-\left(\frac{3}{4}\right)^{n}\gt\frac{2}{3} \\ \left(\frac{3}{4}\right)^{n}\lt\frac{1}{3} \Rightarrow n=4,5,6... \end{array} $$ Hence, the man must fire at least 4 times.