In the image below, I've shared the solution to one of my exams for a course I'm following. My question is regarding the "approximately equal" step on the 2nd and 3rd calculations. Is there a rule of thumb to approximate a sum of binomial distributions? Such as in the 2nd step, where a sum of i=3 to 14 is approximated by just i=3 disregarding the 1-p probability?
The same goes for the 2nd equation, a single step calculates the sum of i=1 to 8 by just taking i=1 and disregarding the 1-p again.
it is just a "make sense rule". If you expand the sum you realize that the first addend is $\approx 2.3\cdot 10^{-8}$ that is quite zero itself and the rest of the addends are very very lower...thus it makes sense to take the first addend as an approximation of the result
These are the first 3 addends rounded at 12 decimals
$$0.000000023194+0.000000000026+0.000000000000\approx 2.23\cdot 10^{-8}$$