A target is to be destroyed in a bombing exer | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the probability of a bomb hitting the target be $p = \frac{3}{4}$.Thus,the probability of a bomb missing the target is $q = 1 - p = \frac{1}{4

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(C) Let the probability of a bomb hitting the target be $p = \frac{3}{4}$.Thus,the probability of a bomb missing the target is $q = 1 - p = \frac{1}{4}$.Let $n$ be the number of bombs dropped. The target is destroyed if there are at least $2$ hits.Let $X$ be the number of hits. We want $P(X \geq 2) \geq 0.99$.This is equivalent to $1 - [P(X = 0) + P(X = 1)] \geq 0.99$.$P(X = 0) + P(X = 1) \leq 0.01 = \frac{1}{100}$.Using the binomial distribution: $P(X = k) = {}^{n}C_{k} p^{k} q^{n-k}$.${}^{n}C_{0} (\frac{3}{4})^{0} (\frac{1}{4})^{n} + {}^{n}C_{1} (\frac{3}{4})^{1} (\frac{1}{4})^{n-1} \leq \frac{1}{100}$.$\frac{1}{4^{n}} + n \cdot \frac{3}{4} \cdot \frac{4}{4^{n}} \leq \frac{1}{100}$.$\frac{1 + 3n}{4^{n}} \leq \frac{1}{100} \Rightarrow 4^{n} \geq 100(3n + 1) = 300n + 100$.For $n = 5$: $4^{5} = 1024$ and $300(5) + 100 = 1600$. ($1024 For $n = 6$: $4^{6} = 4096$ and $300(6) + 100 = 1900$. ($4096 \geq 1900$,true).Thus,the minimum number of bombs is $6$.

Similar Questions

$A$ die is thrown twice. If getting a number greater than $4$ on the die is considered a success,then the variance of the probability distribution of the number of successes is

How many times must a fair coin be tossed so that the probability of getting at least one head is at least $0.8$?

The mean and variance of a random variable $X$ having a binomial distribution are $4$ and $2$ respectively,then $P(X = 1)$ is

The probability of getting a success in a trial is five times that of a failure. The probability of getting at most one success in $5$ trials is:

$A$ fair coin is tossed a fixed number of times. If the probability of getting $5$ heads is equal to the probability of getting $4$ heads,then the probability of getting $6$ heads is

Vedclass Test Series

Exam Paper Generator

Online Exam Module