Question

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

Answer 1

Let $p$ denote the probability of success. Then the probability of failure is $1-p$.
Now $p=2(1-p)$ or $3p=2$ or $p=\frac{2}{3}$, therefore $q=\frac{1}{3}$.
Hence if the number of trials is 6, then the probability of getting atleast 4 success is (by application of Bernoulli trials)
$Pr=\left(\frac{1}{3}{)}^6\right[{}^6{C}_4{2}^4+{}^6{C}_5{2}^5+{}^6{C}_6{2}^6]$

$=\frac{2^4}{3^6}(15+12+4)$

$=\frac{2^4}{3^6}\left(31\right)$ Ans.