Let X be a binomially distributed variate with mean 10 and variance 5. Then P(X > 10) = 
 

$$\begin{gathered} Given mean np=10 \\ and variance npq=5 \\ Now \frac{npq}{np}=\frac{5}{10} \\ \Rightarrow q=\frac{1}{2} \\ Now p=1-q \\ \Rightarrow p=\frac{1}{2} and given n=20 \end{gathered}$$ 

Now $\mathrm{P}(X>10)=\sum _{\mathrm{k}=11}^{20} {}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{k}} {\mathrm{p}}^{\mathrm{k}} {\left(\mathrm{q}\right)}^{\mathrm{n}-\mathrm{k}}$

                           $$\begin{gathered} =\sum _{\mathrm{k}=11}^{20} {}^{20}C_k {\left(\frac{1}{2}\right)}^{\mathrm{k}} {\left(\frac{1}{2}\right)}^{20-\mathrm{k}} \\ ={\left(\frac{1}{2}\right)}^{20} \sum _{\mathrm{k}=11}^{20} {}^{20}C_k \\ =\frac{1}{2^{20}} \sum _{\mathrm{k}=11}^{20} {}^{20}C_k \end{gathered}$$