# 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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