How to solve a binomial inequality

Question

How to solve the following binomial inequality, the variable in this inequality is k, $$\sum\limits_{i=k+1}^{100}{\left( \begin{matrix} 100 \\ i \\ \end{matrix} \right)}{{0.8}^{i}}{{0.2}^{100-i}}\le \alpha,\alpha>0 $$ I think that the following formula is used, but I can not get its result $$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.$$ Please help to suggest me how to find the range of k to satisfy the inequality.

Answer 1

After deeply thinking about this problem, I cannot find any analysis solution for this, because we cannot specify an explicit result of: $$\sum\limits_{i=0}^k{\left( \begin{matrix} 100 \\ i \\ \end{matrix} \right)}{{0.8}^{i}}{{0.2}^{100-i}}$$ I solved this problem by writing a Matlab function to determine k value. It means that I solved it by a numerical method.

Answer 2

$$\sum\limits_{i=k+1}^{100}{\left( \begin{matrix} 100 \\ i \\ \end{matrix} \right)}{{0.8}^{i}}{{0.2}^{100-i}}=\sum\limits_{i=0}^{100}{\left( \begin{matrix} 100 \\ i \\ \end{matrix} \right)}{{0.8}^{i}}{{0.2}^{100-i}}-\sum\limits_{i=0}^k{\left( \begin{matrix} 100 \\ i \\ \end{matrix} \right)}{{0.8}^{i}}{{0.2}^{100-i}}=$$

And now use the binomial expansion you wrote

Answer 3

HINT

Use that

$$\sum\limits_{i=k+1}^{100}...=\sum\limits_{i=0}^{100}...-\sum\limits_{i=0}^{k}...$$