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The Binomial Distribution is a mathematical probability distribution that is used to model the probability of a certain number of successes in a fixed number of independent trials. In Python, the Binomial Distribution can be used to analyze and predict outcomes in various scenarios, such as the success rate of a marketing campaign or the probability of a certain number of defects in a manufacturing process. Python provides various functions and libraries, such as the scipy.stats.binom module, that allow for easy calculation and visualization of the Binomial Distribution. By utilizing these tools, Python can be used to effectively analyze and make informed decisions based on the Binomial Distribution.
Use the Binomial Distribution in Python
The is one of the most commonly used distributions in statistics. It describes the probability of obtaining k successes in n binomial experiments.
If a random variableX follows a binomial distribution, then the probability that X = k successes can be found by the following formula:
P(X=k) = nCk * pk * (1-p)n-k
where:
- n: number of trials
- k: number of successes
- p: probability of success on a given trial
- nCk: the number of ways to obtain k successes in n trials
This tutorial explains how to use the binomial distribution in Python.
How to Generate a Binomial Distribution
You can generate an array of values that follow a binomial distribution by using the function from the numpy library:
from numpy import random #generate an array of 10 values that follow a binomial distribution random.binomial(n=10, p=.25, size=10) array([5, 2, 1, 3, 3, 3, 2, 2, 1, 4])
Each number in the resulting array represents the number of “successes” experienced during 10 trials where the probability of success in a given trial was .25.
How to Calculate Probabilities Using a Binomial Distribution
You can also answer questions about binomial probabilities by using the from the scipy library.
Question 1: Nathan makes 60% of his free-throw attempts. If he shoots 12 free throws, what is the probability that he makes exactly 10?
from scipy.stats import binom #calculate binomial probability binom.pmf(k=10, n=12, p=0.6) 0.0639
The probability that Nathan makes exactly 10 free throws is 0.0639.
Question 2: Marty flips a fair coin 5 times. What is the probability that the coin lands on heads 2 times or fewer?
from scipy.stats import binom #calculate binomial probability binom.cdf(k=2, n=5, p=0.5) 0.5
Question 3: It is known that 70% of individuals support a certain law. If 10 individuals are randomly selected, what is the probability that between 4 and 6 of them support the law?
from scipy.stats import binom #calculate binomial probability binom.cdf(k=6, n=10, p=0.7) - binom.cdf(k=3, n=10, p=0.7) 0.3398
The probability that between 4 and 6 of the randomly selected individuals support the law is 0.3398.
How to Visualize a Binomial Distribution
You can visualize a binomial distribution in Python by using the seaborn and matplotlib libraries:
from numpy import random import matplotlib.pyplot as plt import seaborn as sns x = random.binomial(n=10, p=0.5, size=1000) sns.distplot(x, hist=True, kde=False) plt.show()
The x-axis describes the number of successes during 10 trials and the y-axis displays the number of times each number of successes occurred during 1,000 experiments.