When rolling a die 20 times, let $X$ be the random variable representing the number of times the face 1 appears, and when tossing a coin $n$ times, let $Y$ be the random variable representing the number of times heads appears. Find the minimum value of $n$ such that the variance of $Y$ is greater than the variance of $X$. [4 points]

A random variable $X$ follows a binomial distribution $\mathrm { B } ( 200 , p )$ and the mean of $X$ is 40. What is the variance of $X$? [2 points]
(1) 32
(2) 33
(3) 34
(4) 35
(5) 36

A random variable $X$ follows a binomial distribution $\mathrm{B}(n, p)$. If the mean and standard deviation of the random variable $2X - 5$ are 175 and 12, respectively, what is the value of $n$? [3 points]
(1) 130
(2) 135
(3) 140
(4) 145
(5) 150

A random variable $X$ follows a binomial distribution $\mathrm { B } ( 9 , p )$, and $\{ \mathrm { E } ( X ) \} ^ { 2 } = \mathrm { V } ( X )$. What is the value of $p$? (Here, $0 < p < 1$) [3 points]
(1) $\frac { 1 } { 13 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 11 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 9 }$

A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 3 X ) = 40$. Find the value of $n$. [3 points]

When the random variable $X$ follows the binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 2 } \right)$ and satisfies $\mathrm { E } \left( X ^ { 2 } \right) = \mathrm { V } ( X ) + 25$, what is the value of $n$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 2 X ) = 40$. What is the value of $n$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

Each member of a certain group uses mobile payment with probability $p$. The payment methods of each member are independent. Let $X$ be the number of people among 10 members of the group who use mobile payment. If $D(X) = 2.4$ and $P ( X = 4 ) < P ( X = 6 )$, then $p =$
A. 0.7
B. 0.6
C. 0.4
D. 0.3

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is:
(1) $\frac { 33 } { 2 ^ { 32 } }$
(2) $\frac { 33 } { 2 ^ { 29 } }$
(3) $\frac { 33 } { 2 ^ { 28 } }$
(4) $\frac { 33 } { 2 ^ { 27 } }$

The mean and variance of a binomial distribution are $\alpha$ and $\frac { \alpha } { 3 }$ respectively. If $P ( X = 1 ) = \frac { 4 } { 243 }$, then $P ( X = 4$ or $5 )$ is equal to:
(1) $\frac { 5 } { 9 }$
(2) $\frac { 64 } { 81 }$
(3) $\frac { 16 } { 27 }$
(4) $\frac { 145 } { 243 }$

Let $X$ have a binomial distribution $B ( n , p )$ such that the sum and the product of the mean and variance of $X$ are 24 and 128 respectively. If $P ( X > n - 3 ) = \frac { k } { 2 ^ { n } }$, then $k$ is equal to
(1) 528
(2) 529
(3) 629
(4) 630

In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then $6(n + p - q)$ is equal to:
(1) 51
(2) 52
(3) 53
(4) 50