Question

Consider 5 independent Bernoulli 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. $(\frac{1}{2}\frac{3}{4}]$
• B. $(\frac{3}{4}\frac{11}{12}]$
• C. $[0,\frac{1}{2}]$
• D. $(\frac{11}{12},1]$

Answer 1

Answer: (C)

$[0,\frac{1}{2}]$

$p$(probability of a success)

Probability of at least one failure = 1 - (Probability of no failure)
For 5 events, probability of no failure $=p^5$
Hence, $1-p^5\ge \frac{31}{32}$
$\Rightarrow p^5\le (\frac{1}{2}{)}^5$
$\Rightarrow p\le \frac{1}{2}$
But $p\ge 0$
So, $P$ lies in the interval $[0,\frac{1}{2}]$.