Question

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

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

Answer 1

Answer: (D)

Probability of atleast one failure
$=1-$ no failure $\ge \frac{31}{32}$
$\Rightarrow 1-p^5\ge \frac{31}{32}$
$\Rightarrow p^5\le \frac{1}{32}$
$\Rightarrow p\le \frac{1}{2}$
Also $p\ge 0$
Hence $p\in \left[0,\frac{1}{2}\right]$