An experiment succeeds twice as often as it fails. The probability that in the next six trials there are at least 4 success, is

$\frac{496}{729}$

Let p be the probability of success in a trial.

We have, $p= 2(1 - p)⇒ p=\frac{2}{3}$

The probability of r success in six trials is given by

$P(X=r) = {^6C}_r \left(\frac{2}{3}\right)^r\left(\frac{1}{3}\right)^{6-r}= {^6C}_r2^r \left(\frac{1}{3}\right)^6$

∴ Required probability $= P(X ≥ )$

$=P(X=4) +P(X=5)+P(X=6)$

$= \left(\frac{1}{3}\right)^6 ({^6C}_r × 2^4 +{^6C}_5 × 2^5 + {^6C}_6 × 2^6)=\frac{496}{729}$