The probability that a missile hits a target successfully is $0.75.$ In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than $0.95,$ is

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Solution

Let $n$ be the number of missiles.
$P (completely destroying the target) \geq 0.95$
$\Rightarrow 1 - [^{n} C_{0} {(\frac{1}{4})}^{n} +^{n} C_{1} {(\frac{1}{4})}^{n - 1} (\frac{3}{4}) +^{n} C_{2} {(\frac{1}{4})}^{n - 2} {(\frac{3}{4})}^{2}] \geq 0.95$
$\Rightarrow 1 - [{(\frac{1}{4})}^{n} + n {(\frac{1}{4})}^{n - 1} (\frac{3}{4}) + \frac{n (n - 1)}{2} {(\frac{1}{4})}^{n - 2} {(\frac{3}{4})}^{2}] \geq 0.95$
$\Rightarrow 1 - 0.95 \geq {(\frac{1}{4})}^{n} [\frac{9 n^{2} - 3 n + 2}{2}]$
$\Rightarrow 9 n^{2} - 3 n + 2 \leq \frac{4^{n}}{10}$
The least value of $n$ is $6.$

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The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is

Let n be the number of missiles. P(completely destroying the target)≥0.95 ⇒1−[nC0(14)n+nC1(14)n−1(34)+nC2(14)n−2(34)2]≥0.95 ⇒1−[(14)n+n(14)n−1(34)+n(n−1)2(14)n−2(34)2]≥0.95 ⇒1−0.95≥(14)n[9n2−3n+22] ⇒9n2−3n+2≤4n10 The least value of n is 6.