Question

In a certain experiment the probability of success is twice the probability of faliure. Find the probaility of at least four successes in six trails.

• A. $\frac{297}{729}$
• B. $\frac{233}{729}$
• C. $\frac{432}{729}$
• D. $\frac{496}{729}$

Answer 1

Answer: (D)

$\frac{496}{729}$

Let p is the probability of success and q the probability of failure .
Then $p+q=1$
Given that $p=2q$
$\Rightarrow 2q+q=1$
$q=\frac{1}{3}$
$\Rightarrow p=\frac{2}{3}$
Now the probability of at least four successes in six trials
$={}^6{C}_4{p}^4{q}^2{+}^6{C}_5{p}^{5}q{+}^6{C}_6{p}^6$
$=15{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2+6{\left(\frac{2}{3}\right)}^5\left(\frac{1}{3}\right)+{\left(\frac{2}{3}\right)}^6$
$=\frac{1}{3^6}[240+192+64]=\frac{496}{729}$