If the sum and the product of the mean and va | vedclass

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

Question

(C) For a binomial distribution,the mean is $\mu = np$ and the variance is $\sigma^2 = npq$,where $p+q=1$.Given $np + npq = 24$ and $(np)(npq) = 128$.

Vedclass · Accepted Answer

$\frac{33}{2^{28}}$

Vedclass · Answer

(C) For a binomial distribution,the mean is $\mu = np$ and the variance is $\sigma^2 = npq$,where $p+q=1$.Given $np + npq = 24$ and $(np)(npq) = 128$.Let $A = np$ and $B = npq$. Then $A+B=24$ and $AB=128$.The quadratic equation $t^2 - 24t + 128 = 0$ has roots $t = 8$ and $t = 16$.Case $1$: $np = 16$ and $npq = 8$. Then $q = \frac{8}{16} = \frac{1}{2}$,so $p = \frac{1}{2}$. Thus $n(\frac{1}{2}) = 16 \implies n = 32$.Case $2$: $np = 8$ and $npq = 16$. Then $q = \frac{16}{8} = 2$,which is impossible since $q \leq 1$.So,$n=32, p=\frac{1}{2}, q=\frac{1}{2}$.The probability of one or two successes is $P(X=1) + P(X=2) = {}^{32}C_1 (\frac{1}{2})^1 (\frac{1}{2})^{31} + {}^{32}C_2 (\frac{1}{2})^2 (\frac{1}{2})^{30}$.$= 32 \cdot (\frac{1}{2})^{32} + \frac{32 \cdot 31}{2} \cdot (\frac{1}{2})^{32} = (32 + 496) \cdot \frac{1}{2^{32}} = 528 \cdot \frac{1}{2^{32}} = \frac{33 \cdot 16}{2^{32}} = \frac{33}{2^{28}}$.

If the sum and the product of the mean and variance of a binomial distribution are $24$ and $128$ respectively,then the probability of one or two successes is:

Similar Questions

$A$ person fails $4$ times in a game when he plays $9$ times. If he plays $15$ times, the probability of having success at most one 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

In a Binomial distribution $B(n, p)$,the sum and product of the mean and the variance are $5$ and $6$ respectively,then $6(n+p-q)=$

$A$ fair coin is tossed $100$ times. The chance of getting a head an even number of times is

$A$ fair die with faces $1, 2, 3, 4, 5,$ and $6$ is tossed $4$ times. What is the probability that in all $4$ tosses,the value obtained is not less than $2$ and not greater than $5$ (in $/81$)?

Vedclass Test Series

Exam Paper Generator

Online Exam Module