Question

Consider $5$ independent Bernoulli trials each with the probability of success $p$. If the probability of at least one failure is greater than or equal to$\dfrac{{31}}{{32}}$, then $p$ lies in the interval:
A. $(\dfrac{1}{2},\dfrac{3}{4}]$
B. $(\dfrac{3}{4},\dfrac{{11}}{{12}}]$
C. $\left[ {0,\dfrac{1}{2}} \right]$
D. $(\dfrac{{11}}{{12}},1]$

Answer 1

In this question, we are given that probability of at least one failure is greater than or equal to$\dfrac{{31}}{{32}}$. So, we solve this equation that probability of at least one failure $\geqslant \dfrac{{31}}{{32}}$, using definition of Bernoulli trials.Bernoulli trial is the experiment with exactly two possible outcomes, success and failure and using basic probability that${\text{probability of occurrence of event A at least once = 1 - (probability of no occurrence of event A)}}$, for any event $A$.We will try to find a value of $p$ from it.

Complete step-by-step answer:
In this question, we are given,
$5$ independent Bernoulli trials each with the probability of success$p$.
We know that Bernoulli trial is the experiment with exactly two possible outcomes, success and failure. Now further we are given that the probability of at least one failure is greater than or equal to$\dfrac{{31}}{{32}}$.
So, let the probability of at least one failure be $q$
So, we get that $q \geqslant \dfrac{{31}}{{32}}$ $- - - - (1)$
Now we can write that
${\text{probability of occurrence of event A at least once = 1 - (probability of no occurrence of event A)}}$, for any event $A$. $- - - - (2)$
Now for using (2) for occurrence of failure, we get,
${\text{probability of at least one failure = 1 - (probability of no failure)}}$ $- - - - - - (3)$
Now probability of no failure means that every time we get success and we know that there are a total of 5 independent trials. So,
Probability of no failure$= p.p.p.p.p = {p^5}$ $- - - - - - (4)$
Now substituting the value of probability of no failure from (4) in (3), we get
${\text{probability of at least one failure(q) = 1 - }}{p^5}$
$\Rightarrow q = 1 - {p^5}$ $- - - - - (5)$
Now substituting the value of $q$ from (5) in (1), we get
$1 - {p^5} \geqslant \dfrac{{31}}{{32}}$
${p^5} \leqslant 1 - \dfrac{{31}}{{32}}$
Now solving it further, we get,
${p^5} \leqslant \dfrac{1}{{32}}$
${p^5} \leqslant \dfrac{{32 - 31}}{{32}}$
$p \leqslant \dfrac{1}{2}$ $- - - - (5)$
Now we know that if $P(A)$ is the probability of any event$A$, then $0 \leqslant P(A) \leqslant 1$
Hence as $p \leqslant \dfrac{1}{2}$, so using $0 \leqslant P(A) \leqslant 1$, where $p$ is probability for success-
$\Rightarrow 0 \leqslant p \leqslant \dfrac{1}{2}$
So, $p \in \left[ {0,\dfrac{1}{2}} \right]$

So, the correct answer is “Option C”.

Note: In these kind of questions, we should remember that the experiment whose outcomes are exactly of two types that is success or failure is known as the Bernoulli trial.If $q$ is the probability of the failure and $p$ is the of success, then $q = 1 - p$.