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 \(31\over 32\), then p lies in the interval

Solution :

Here, n = 5 and r \(\ge\) 1

\(\therefore\)   p(X = r) = \(^nC_r\) \(p^{n-r}\) \(q^r\)

P(X \(\ge\) 1) = 1 – P(X = 0)

= 1 – \(^5C_0 . p^5 . q^0\) \(\ge\) \(31\over 32\)   [Given]

\(\implies\)   \(p^5\) \(\le\) 1 – \(31\over 32\) = \(1\over 32\)

\(\therefore\)  p \(\le\) \(1\over 2\) and p \(\ge\) 0

\(\implies\)  p \(\in\)  [0, 1/2]


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