Consider the following typical probability scenario:

Probability Scenario

I defined this function to handle that scenario, I'm curious if Python has a more efficient method to handle this, or if this is the best way:

from scipy.special import binom

def grade_of_service(n, p, c):
    prob = 0
    k = c + 1
    while k <= n:
        prob += binom(n, k)*(p**k)*((1-p)**(n-k))
        k += 1
    return prob

EDIT: Here's an example solution from the author of this example: "If n = 100, p=0.1, and c=15, the probability of interest turns out to be 0.0399." My probability does round to this result. But due to the roundoff error / numerical precision, it doesn't seem to matter if k = n or k = n+1.

  • \$\begingroup\$ Do you have some example inputs we can test with? Preferably edge cases. \$\endgroup\$
    – Peilonrayz
    Jun 21, 2018 at 22:05
  • \$\begingroup\$ @Peilonrayz, I edited to provide an example solution from the example author. I don't have any edge cases yet but will mess with some numbers to see if I can find some. \$\endgroup\$
    – Hanzy
    Jun 21, 2018 at 22:11

1 Answer 1


Pure Python

  1. You should use a for loop rather than a while loop.
  2. You can use a generator comprehension to build all prob's.
  3. You can use sum to get the sum.
def grade_of_service(n, p, c):
    return sum(binom(n, k)*(p**k)*((1-p)**(n-k)) for k in range(c+1, n+1))

Numpy and friends

  1. Use numpy.arange rather than range.
  2. Write out the equation same as above, just not in a comprehension.
  3. Change sum to numpy.sum.
import numpy

def grade_of_service(n, p, c):
    k = numpy.arange(c+1, n+1)
    return numpy.sum(binom(n, k)*(p**k)*((1-p)**(n-k)))

This has a problem with numbers that exceed a certain size, as numpy numbers are finite.

  • \$\begingroup\$ Thank you this will be more efficient in the case the variables get incredibly large, the generator will save me a lot of memory usage. In practical use if the probability meets a threshold is really all that matters, so as long as the precision is down to .001 it probably won’t make a difference. Since they are probabilities they will never exceed 1, and anything below .01 in practice probably doesn’t need to be that precise. \$\endgroup\$
    – Hanzy
    Jun 21, 2018 at 22:17

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