Let p denotes the probability of success and q denotes failure.
An experiment succeeds twice as often as it fails
$$
\begin{aligned}
&\Rightarrow \quad \mathrm{p}=2 \mathrm{q}=2(1-\mathrm{p})=2-2 \mathrm{p} \\
&\therefore \quad 3 \mathrm{p}=2 \quad \text { or } \mathrm{p}=\frac{2}{3} \quad \therefore \mathrm{q}=\frac{1}{3}
\end{aligned}
$$
Probability that in six trails, there are atlest 4 successes
$$
\begin{aligned}
&=\mathrm{P}(4)+\mathrm{P}(5)+\mathrm{P}(6)={ }^6 \mathrm{C}_4 \mathrm{q}^2 \mathrm{p}^4+{ }^6 \mathrm{C}_5 \mathrm{qp}^5+\mathrm{p}^6 \\
&={ }^6 \mathrm{C}_2\left(\frac{1}{3}\right)^2\left(\frac{2}{3}\right)^4+{ }^6 \mathrm{C}_1\left(\frac{1}{3}\left(\frac{2}{3}\right)^5+\left(\frac{2}{3}\right)^6\right. \\
&=\frac{6 \times 5}{2} \times \frac{16}{3^6}+\frac{6 \times 32}{3^6}+\frac{64}{3^6} \\
&=\frac{15 \times 16+6 \times 32+64}{3^6}=\frac{496}{729}=\frac{31}{9}\left(\frac{2}{3}\right)^4
\end{aligned}
$$