This is from Feller's Introduction to Probability Theory and Its Applications: we have the equation: $$b(k;n,p)/b(k−1;n,p) =(n−k+1)*p/kq=1+ ((n+1)p−k)/kq)$$ and b(k;n,p) is greater than the preceding one for $k<(n+1)*p$ and is smaller for $k>(n+1)*p$ and from that we have got the inequality: $$(n+1)*p - 1<m<=(n+1)*p$$ where $m$ is "the most probable number of successes. I have understood the idea, but i can't figure out how we got the last equation?Can you explain me this step by step?
And also one more question: I have completely lost in definiton of upper and lower bound of "probability at least r successes". Can you explain me this also, or should i create separate question? Thanks
The expression you received for the ratio in the first expression is of the form 1+something. As long as this something is positive you get an increasing probability and once "something" becomes negative the probability starts decreasing as $k$ increases. Hence the most probable number of results is the maximal $k$ for which "something" is still positive.