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 $r \geq 1$
$\therefore P(x=r)={ }^n C_r p^{n-r} q^r$
$P(x \geq 1)=1-P(x=0)$
$=1-{ }^5 C_0 \cdot p^5 \cdot q^0 \geq \frac{31}{32}$
$\Rightarrow p^5 \leq 1-\frac{31}{32}=\frac{1}{32}$
$\therefore p \leq \frac{1}{2}$ and $p \geq 0$
$\Rightarrow p \in\left[0, \frac{1}{2}\right]$