My question is relating finding the probability using Normal Approximation.
Let X be a random variable with P(X=-1)=P(X=0)=0.25 and P(X=1)=0.5. Let S be the sum of 25 independent random variables, each with the same distribution as X.
Calculate (a)P(S>0) (b)P(S<0) (c)P(S=0)
I know how to do normal approximation, and also have the solutions bank for this question. But I don't know how to find the Standard Deviation. Mean is np , hence 25*0.25=6.25. The solutions bank states the standard deviation is 5/4root11. I calculate it as root(25*0.25*0.75) but I know its wrong too.
You have a typo in your question saying $P(X=-1)$ twice.
You may have intended $P(X=-1)=P(X=0)=0.25$ and $P(X=1)=0.5$ which would make $E[X]=\frac14$ and so multiplying by $25$ you get $E[S]=\frac{25}{4}$
If so, $\text{Var}(X) = \frac14(-1-\frac14)^2+\frac14(0-\frac14)^2+\frac12(1-\frac14)^2 =\frac{11}{16}$ and so multiplying by $25$ you get $\text{Var}(S)=\frac{25\times 11}{16}$ the square root of which is $\frac54\sqrt{11}$