If the difference between the mean and varian | vedclass

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

Question

(A) For a binomial distribution,Mean $= np$ and Variance $= npq$,where $n=5$ is the number of trials,$p$ is the probability of success,and $q=1-p$ is

Vedclass · Accepted Answer

$\frac{80}{243}$

Vedclass · Answer

(A) For a binomial distribution,Mean $= np$ and Variance $= npq$,where $n=5$ is the number of trials,$p$ is the probability of success,and $q=1-p$ is the probability of failure. Given that the difference between mean and variance is $\frac{5}{9}$: $np - npq = \frac{5}{9}$ $np(1-q) = \frac{5}{9}$ Since $1-q = p$,we have $np^2 = \frac{5}{9}$. Substituting $n=5$: $5p^2 = \frac{5}{9} \Rightarrow p^2 = \frac{1}{9} \Rightarrow p = \frac{1}{3}$. Then $q = 1 - \frac{1}{3} = \frac{2}{3}$. The probability of getting $2$ successes in $5$ trials is given by the binomial formula $P(X=k) = {^nC_k} p^k q^{n-k}$: $P(X=2) = {^5C_2} 	imes (\frac{1}{3})^2 	imes (\frac{2}{3})^{5-2}$ $P(X=2) = 10 	imes \frac{1}{9} 	imes \frac{8}{27} = \frac{80}{243}$.

If the difference between the mean and variance of a binomial variate is $\frac{5}{9}$,then the probability of getting $2$ successes for an event when the experiment is conducted $5$ times,is

Similar Questions

In an examination,$20$ questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads,he answers 'true'; if it falls tails,he answers 'false'. Find the probability that he answers at least $12$ questions correctly.

The probability that a person wins a prize on a lottery ticket is $\frac{1}{4}$. If he purchases $5$ lottery tickets at random,then the probability that he wins at least one prize is

$A$ box contains $30$ toys of the same size,in which $10$ toys are white and the remaining toys are blue. $A$ toy is drawn at random from the box and it is replaced in the box after noting down its colour. If $5$ toys are drawn in this way,then the probability of getting at most $2$ white toys is

The mean and the variance of a binomial distribution are $4$ and $2$ respectively. Then the probability of $2$ successes is

Vedclass Test Series

Exam Paper Generator

Online Exam Module