A random experiment is conducted five times. | vedclass

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

Question

(A) Given that the experiment is conducted $n = 5$ times. Let $p$ be the probability of success and $q = 1 - p$ be the probability of failure. The mea

Vedclass · Accepted Answer

$\frac{64}{81}$

Vedclass · Answer

(A) Given that the experiment is conducted $n = 5$ times. Let $p$ be the probability of success and $q = 1 - p$ be the probability of failure. The mean of a binomial distribution is $np$ and the variance is $npq$. Given $np - npq = \frac{5}{9}$. Substituting $n = 5$: $5p - 5pq = \frac{5}{9} \implies p - pq = \frac{1}{9}$. Since $1 - q = p$,we have $p(1 - q) = p^2 = \frac{1}{9}$,which gives $p = \frac{1}{3}$. Thus,$q = 1 - \frac{1}{3} = \frac{2}{3}$. The probability of getting at most two successes is $P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$. Using the formula $P(X = k) = {}^nC_k p^k q^{n-k}$: $P(X = 0) = {}^5C_0 (\frac{1}{3})^0 (\frac{2}{3})^5 = 1 	imes 1 	imes \frac{32}{243} = \frac{32}{243}$. $P(X = 1) = {}^5C_1 (\frac{1}{3})^1 (\frac{2}{3})^4 = 5 	imes \frac{1}{3} 	imes \frac{16}{81} = \frac{80}{243}$. $P(X = 2) = {}^5C_2 (\frac{1}{3})^2 (\frac{2}{3})^3 = 10 	imes \frac{1}{9} 	imes \frac{8}{27} = \frac{80}{243}$. Summing these probabilities: $P(X \leq 2) = \frac{32}{243} + \frac{80}{243} + \frac{80}{243} = \frac{192}{243} = \frac{64}{81}$.

$A$ random experiment is conducted five times. If the number of successes of the experiment follows a binomial distribution such that the difference between the mean and variance of the successes is $\frac{5}{9}$,then the probability of getting at most two successes is

Similar Questions

If the sum of mean and variance of a binomial distribution for $5$ trials is $1.8$,then the probability of success is:

If $X$ has a binomial distribution with mean $np$ and variance $npq$,then $\frac{P(X = k)}{P(X = k - 1)}$ is

What is the probability of getting at least $1$ tail when a coin is tossed $4$ times?

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least $90\%$ is:

If the probability that a student is not a swimmer is $\frac{1}{5}$,then the probability that out of $5$ students one is a swimmer is

Vedclass Test Series

Exam Paper Generator

Online Exam Module