Recall that for [tex]X\sim\mathrm{Bin}(n,p)[/tex], i.e. a random variable [tex]X[/tex] following a binomial distribution over [tex]n[/tex] trials and with probability parameter [tex]p[/tex],
[tex]\mathbb P(X=x)=f_X(x)=\begin{cases}\dbinom nx p^x(1-p)^{n-x}&\text{for }x\in\{0,1,\ldots,n\}\\\\0&\text{otherwise}\end{cases}[/tex]
So you have
[tex]\mathbb P(X=2)=\dbinom52\left(\dfrac14\right)^2\left(1-\dfrac34\right)^{5-2}=\dfrac{135}{512}\approx0.26[/tex]
[tex]\mathbb P(2\le X\le4)=\displaystyle\sum_{x=2}^4\mathbb P(X=x)[/tex]
[tex]=\dbinom52\left(\dfrac14\right)^2\left(1-\dfrac14\right)^{5-2}+\dbinom52\left(\dfrac14\right)^3\left(1-\dfrac14\right)^{5-3}+\dbinom52\left(\dfrac14\right)^4\left(1-\dfrac14\right)^{5-4}[/tex]
[tex]=\dfrac{375}{1024}\approx0.37[/tex]
The expected value of [tex]X\sim\mathrm{Bin}(n,p)[/tex] is simply [tex]np[/tex], while the standard deviation is [tex]\sqrt{np(1-p)}[/tex]. In this case, they are [tex]\dfrac54=1.25[/tex] and [tex]\sqrt{\dfrac{15}{16}}\approx0.97[/tex], respectively.
