The number of trials conducted in a binomial | vedclass

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

Question

(D) For a binomial distribution,the mean is $\mu = np$ and the variance is $\sigma^2 = npq$,where $q = 1 - p$.Given $n = 6$ and $\mu - \sigma^2 = \fra

Vedclass · Accepted Answer

$\frac{154}{4^6}$

Vedclass · Answer

(D) For a binomial distribution,the mean is $\mu = np$ and the variance is $\sigma^2 = npq$,where $q = 1 - p$.Given $n = 6$ and $\mu - \sigma^2 = \frac{27}{8}$.Substituting the formulas: $np - npq = \frac{27}{8} \implies np(1 - q) = \frac{27}{8}$.Since $1 - q = p$,we have $np^2 = \frac{27}{8}$.Substituting $n = 6$: $6p^2 = \frac{27}{8} \implies p^2 = \frac{27}{48} = \frac{9}{16}$.Thus,$p = \frac{3}{4}$ and $q = 1 - \frac{3}{4} = \frac{1}{4}$.The probability of getting $X$ successes is given by $P(X = k) = \binom{n}{k} p^k q^{n-k}$.We need to find $P(X \le 2) = P(X = 0) + P(X = 1) + P(X = 2)$.$P(X = 0) = \binom{6}{0} (\frac{3}{4})^0 (\frac{1}{4})^6 = 1 	imes 1 	imes \frac{1}{4^6} = \frac{1}{4^6}$.$P(X = 1) = \binom{6}{1} (\frac{3}{4})^1 (\frac{1}{4})^5 = 6 	imes \frac{3}{4^6} = \frac{18}{4^6}$.$P(X = 2) = \binom{6}{2} (\frac{3}{4})^2 (\frac{1}{4})^4 = 15 	imes \frac{9}{4^6} = \frac{135}{4^6}$.Summing these: $P(X \le 2) = \frac{1 + 18 + 135}{4^6} = \frac{154}{4^6}$.

The number of trials conducted in a binomial distribution is $n = 6$. If the difference between the mean and variance of this variate is $\frac{27}{8}$,then the probability of getting at most $2$ successes is:

Similar Questions

When two dice are rolled, let $x$ be the probability that the sum of the numbers appearing on the dice is at most $7$. Let $y$ be the probability of getting a sum of $7$ at least once when a pair of dice are rolled $n$ times. In order to have $y > x$, the minimum value of $n$ is

The mean of a binomial variate $X \sim B(n, p)$ is $1$. If $n > 2$ and $P(X=2)=\frac{27}{128}$, then the variance of the distribution is

The probability of guessing correctly at least $7$ out of $10$ answers in a "True" or "False" test is = $ . . . . . . $

If the mean and the variance of a binomial variate $X$ are $1$ and $0.75$ respectively, then which of the following is true?

If the mean and standard deviation of a binomial distribution are $20$ and $4$,respectively,then the number of trials is $.......$

Vedclass Test Series

Exam Paper Generator

Online Exam Module