Test Index

Probability Part 1

Section: Mathematics

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Question : 79 of 100

Marks: +1, -0

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}

p

lies in the interval

$(\;\frac{3}{4}, \;\frac{11}{12}]$
$\left[0, \;\frac{1}{2}\right]$
$(\;\frac{11}{12}, 1]$
$(\;\frac{1}{2}, \;\frac{3}{4}]$

Solution:

Here is 5 trials.

So according to Bernoulli trial

n=5

P(

at least one failure

)=1-P(

no failure

)

According to question,

1-P(

no failure

) \geq \;\frac{31}{32}

\Rightarrow 1-P(x=5) \geq \;\frac{31}{32}

[Note: no failure = all success]

\;\Rightarrow 1-{}^5C_5 p^5 \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}

\;\therefore p \in \left[0, \;\frac{1}{2}\right]

Note:

p

can not be less than 0 as probability is always

\geq 0

and

\leq 1

.

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