Question

An experiment succeeds twice as often as it fails. Find the chance that in the next six trials, there shall be at least four successes.

• A. $\frac{233}{729}$
• B. $\frac{64}{729}$
• C. $\frac{496}{729}$
• D. $\frac{432}{729}$

Answer 1

Answer: (C)

$\frac{496}{729}$

Let $p$ denotes the probability of success and $q$ denotes the probability of failure.
Then $p+q=1$
Given $p=2q$.
Thus $p=\frac{2}{3},q=\frac{1}{3}.$
Hence probability of at least four successes in the next six trials is ${=}^6{C}_4{p}^4{q}^2{+}^6{C}_5{p}^{5}q{+}^6{C}_6{p}^6$
$=15{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2+6{\left(\frac{2}{3}\right)}^5\left(\frac{1}{3}\right)+{\left(\frac{2}{3}\right)}^6$
$=\frac{426}{729}$