Binomial Probability Using Python
I enjoy programming. I really like the applications of programming in areas of finance and Operations.
Questions Answered
We discuss in brief what is a binomial event, how binomial probability works and then go on to program binomial probability in Python. We will also solve the following interesting binomial problem.
Problem: The ratio of boys to girls for babies born in a country is 1.09:1. What proportion of its families with exactly 6 children will have at least 3 boys?
Assumption: There are only 2 possible genders.
Binomial Event (Bernoulli Trial)
A binomial event is a statistical experiment that has the following properties:
- The experiment consists of n repeated trials. e.g. 10 throws of a coin.
- The trials are independent of each other. e.g. Current outcome doesn't affect the next
- The outcome of each trial is either success () or failure (). o.r. Head or Tail.
- Probability of success of 1 trial is p and that of failure is 1-p. i.e. total probability is 1.
Ingredients for Coding in Python
We need:
- from statistics Import math to call builtin factorial function.
- Write a function for calculating the binomial distribution
- That's it.
Python Code
# Import stat module for factorial from statistics import math # Define function to get binomial distribution. # Here x in the number of successful outcomes # n is number of trials # p is probability of success. # example:binom_dist(2, 5, 1.09) gives probability of 2 boys # in a family of 5 in a country where # male to female ratio at birth is 1.09:1 def binom_dist(x, n, p): success_prob = 0 success_prob =(math.factorial(n)/(math.factorial(x)*math.factorial(n-x)))*(p**x)*((1-p)**(n-x)) return boys
Using this code to solve problems
Building on the child birth example. Let's assume that male:female ratio is 1.1.
Problem1: What is the probability that a family of 2 has exactly 1 boy and 1 girl.
print(binom_dist(1,2,1.1)) output: 0.773
Problem2: What is the probability that a family of 6 has at least 3 girls.
This is a little tricky one. At least 3 girls means cases where girls' number is 3,4,5 &6. Now, we need to calculate cumulative probability.
# Since we need prob at multiple points we have to loop it in # We declare a cum_prob variable # We use 1/1.1 as prob of success as we are using it for girls # n=6 p=1/1.1 cum_prob=0 for i in range(3,n+1,1): cum_prob+=run_binom(i, n, p) print(cum_prob) #output: 0.611
Bare code
from statistics import math
def binom_dist(x, n, p):
boys = 0
boys =(math.factorial(n)/(math.factorial(x)*math.factorial(n-x)))*(p**x)*((1-p)**(n-x))
return boys
#Problem1: What is the probability that a family of 2 has exactly 1 boy and 1 girl.
print(binom_dist(1,2,1.1))
#output: 0.773
#Problem2: What is the probability that a family of 6 has at least 3 girls.
# Since we need prob at multiple points we have to loop it in
# We declare a cum_prob variable
# We use 1/1.1 as prob of success as we are using it for girls
#
n=6
p=1/1.1
cum_prob=0
for i in range(3,n+1,1):
cum_prob+=run_binom(i, n, p)
print(cum_prob)
#output: 0.611
