Hi All,

In my textbook there is the following binomial expression which is solved to a numerical answer:

$\displaystyle \sum_{i=13}^{25} {25 \choose i} (0.45)^{i} (0.55)^{25-i} \approx 0.306$

I'd like to know if there is a way to calculate this result without having to calculate the entire summation.

I know that from the binomial theorem:

$\displaystyle \sum_{k=0}^{n} {n \choose k} x^{n-k} y^{k} = (x+y)^{n}$

But how can I use this to calculate the value above? Because the lower bound of the sum in the textbook example is not 0, I cannot us the binomial theorem straight away. Is there some way in which I can rewrite the formula to make use of the binomial theory?

Kind regards,

Chris