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 $\geq \frac{31}{32}$
$\Rightarrow 1-p^{5} \geq \frac{31}{32}$
$\Rightarrow p^{5} \leq \frac{1}{32}$
$\Rightarrow p \leq \frac{1}{2}$
Also $p \geq 0$
Hence $p \in\left[0, \frac{1}{2}\right]$