An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.

.: Let $p$ be the probability of success and $q$ be the probability of failure.

We are given, $p = 2q$

We know, $p + q = 1$

On solving,

we get $p = \cfrac{2}{3}$ and $q = \cfrac{1}{3}$

$X$ has a binomial distribution with $n = 6,p = \cfrac{2}{3},q = \cfrac{1}{3}$

$\therefore$ $P(X = r){ = ^n}{C_r}{(q)^{n - r}}{p^r}$

$\Rightarrow$ Required probability $= P$

(atleast 4 successes in next 6 trials)

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

$= 15{\left( {\cfrac{1}{3}} \right)^2}{\left( {\cfrac{2}{3}} \right)^4} + 6\left( {\cfrac{1}{3}} \right){\left( {\cfrac{2}{3}} \right)^5} + (1)(1){\left( {\cfrac{2}{3}} \right)^6}$

$= {\left( {\cfrac{2}{3}} \right)^4}\left[ {\cfrac{{15}}{9} + \cfrac{4}{3} + \cfrac{4}{9}} \right] = \cfrac{{31}}{9}{\left( {\cfrac{2}{3}} \right)^4}$