Question

Consider 5 independent Bernoulli's trials each with probability of success $p$. If the probability of atleast one failure is greater than or equal to $\frac{31}{32}$, then $p$ lies in the interval

• A. $\left(\frac{3}{4},\frac{11}{12}\right]$
• B. $\left[0,\frac{1}{2}\right]$
• C. $\left(\frac{11}{12},1\right]$
• D. $\left(\frac{1}{2},\frac{3}{4}\right]$

Answer 1

Answer: (B)

$n=5$ and $r\ge 1$
$\therefore P(x=r)={}^n{C}_r{p}^{n-r}{q}^r$
$P(x\ge 1)=1-P(x=0)$
$=1-{}^5{C}_0\cdot p^5\cdot q^0\ge \frac{31}{32}$
$\Rightarrow p^5\le 1-\frac{31}{32}=\frac{1}{32}$
$\therefore p\le \frac{1}{2}$ and $p\ge 0$
$\Rightarrow p\in \left[0,\frac{1}{2}\right]$