What is the formula for the probability of getting r OR MORE (not
exactly) positive events for n tries?

The formula for getting the probability of r events after n tries is:
P(r|n) = (n!/r!(n-r)!)*p^r*q*(n-r), where q = 1.0-p. For the case of q
= p = 0.5 we get the formula (n!/r!(n-r)!)(1/2^n).

But this is the exact number for r events. I want to know the formula
for r or more events (up to n). In other words, there may be more then
r events. What is the total probability to get r OR MORE events given
n tries?

1 - probability of getting (r-1) in n tries.
My reasoning is intuitive rather than mathematical, hence am offering
no explanation. Eager to see how the reasoning for this turns out
Carryou

The answer is equation 3 of the Mathworld article on the binomial
distribution. It is expressed in terms of the incomplete beta
function. For your case, plug (r-1) in for n in that equation.

http://mathworld.wolfram.com/BinomialDistribution.html