How can I get N trials from binomial distribution (Edited)

Question

$$ \sum_{k=0}^{17}{_NC_k}\times 0.1^k\times 0.9^{N-k}<0.004 $$

How can I get a $N$ from above inequality?

Answer 1

As commented by @joriki, in general for this summation a numerical approach is the way to go.

Courtesy of Wolfram Alpha, after some quick trial and error one finds that at $N = 305$ the summation is about 0.00405, and at $N = 306$ the sum is roughly 0.00385.

The minimal $N$ that satisfies your inequality is $N = 306$, which happens to be a multiple of 17.

Whether there's an analytic solution for this particular set of numbers ($17$ and $0.004 = \frac2{500}$ with $p=0.1 = \frac1{10}$) is beyond me, and I'd like to hear from anyone who has an idea.