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 $\frac{31}{32},$ the $p$lies in the interval

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

Answer 1

The correct option is B

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

Find the required interval:

Let $p$ be the probability of success

The probability of at least one failure $=1-$probability of no failure

Hence, for $5$ events the probability of no failure is $p^5$.

Probability of at least one failure $=1–p^5$

Now,

$1-p^5\ge \frac{31}{32}$

$\Rightarrow$$p^5\le {\left(\frac{1}{2}\right)}^5$

$\Rightarrow$ $p\le \frac{1}{2}$

We know that the probability value should be equal to or greater than $0$.

Thus, the value of $p$lies in the interval $[0,\frac{1}{2}]$

Hence, option (B) is correct.