Probability of something happening at least n out of m times

Mathematica, 29 bytes

BetaRegularized[#3,#,1+#2-#]&

Takes input in the order n,m,p. Mathematica is so good, it even golfs your code for you:

BetaRegularized is the regularised incomplete beta function.

R, 32 31 bytes

function(p,n,m)pbeta(p,m,1+n-m)

edit - 1 byte switching to beta distribution (along the lines of @Sp3000 Mathematica Answer)

Python, 57 bytes

f=lambda p,n,m:m and(1-p)*f(p,n,m-1)+p*f(p,n-1,m-1)or n<1

The recursive formula for binomial coefficients, except the base case m==0 indicates whether the remaining number of required successes n is nonnegative, with True/False for 1/0. Because of its exponential recursion tree, this stalls on large inputs.