Python Tutorial

A key idea in probability theory and statistics is the binomial distribution. It explains the likelihood of attaining specific successes in a set number of independent Bernoulli trials, where each trial may only result in success or failure.

Following are the Main Features of the Binomial Distribution:

Fixed Number of Trials (n): The binomial distribution deals with a fixed number of trials or experiments, represented by 'n'. Every hardship stands alone from the others.
Probability of Success (p): A fixed chance of success, indicated by the letter "p," exists for each trial. The failure probability is thus "1 - p."
Independence: It is expected that the trial results are independent. Implies that the result of one trial has no bearing on the outcome of another.
The likelihood of achieving precisely 'k' successes in 'n' trials is given by the probability mass function (PMF) of the binomial distribution:

P(X=k) = (nCk) .p^k. (1-p)^n-k

Where:

P(X=k) is the likelihood of achieving precisely 'k' achievements.
(nCk) is the binomial coefficient shows how many possible methods there are to choose 'k' successes from 'n' trials.
P^k is the likelihood that 'k' will succeed.
(1-p)^n-k is the likelihood of failures of size 'n - k'.

To obtain distribution we have to import some libraries

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm, binom, poisson, expon

Examples 1: Coin Flips

Take a fair coin and toss it ten times. Count how many times it lands heads up. Every flip has two outcomes: heads (success) or tails (failure).

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom
n = 10  # Number of coin flips
p = 0.5  # Probability of heads (success)
k_values = np.arange(0, n + 1)
pmf = binom.pmf(k_values, n, p)
plt.bar(k_values, pmf)
plt.xlabel('Number of Heads')
plt.ylabel('Probability')
plt.title('Binomial Distribution - Coin Flips')
plt.show()

Output:

Example 2: Test pass rates

Each question on a multiple-choice test has four possible answers. A student guesses at random the solutions to a 20-question test. What is the probability that the student will pass if at least ten questions are answered correctly?

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom
n = 20  # Number of questions
p = 0.25  # Probability of correct answer (random guess)
k_values = np.arange(0, n + 1)
pmf = binom.pmf(k_values, n, p)
plt.bar(k_values, pmf)
plt.xlabel('Number of Correct Answers')
plt.ylabel('Probability')
plt.title('Binomial Distribution - Test Pass Rates')
plt.show()

Output:

Example 3: Manufacturing Defects

100 products are randomly selected from a production process, and the quantity of faulty goods is tallied. What is the likelihood of discovering precisely 10 faulty goods if the chance of a defective item is 0.1?

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom

n = 100  # Number of items sampled
p = 0.1  # Probability of a defective item
k_values = np.arange(0, n + 1)
pmf = binom.pmf(k_values, n, p)
plt.bar(k_values, pmf)
plt.xlabel('Number of Defective Items')
plt.ylabel('Probability')
plt.title('Binomial Distribution - Manufacturing Defects')
plt.show()

Example 4: Online Advertising Click-Through Rates

Consider that you are managing an internet advertising campaign and are aware that historically, your adverts have a click-through rate (CTR) of 0.1 (10%). You want to know the chance of receiving a specific number of clicks out of every 100 impressions while running a new advertisement.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom

n = 100  # Number of ad impressions
p = 0.1  # Probability of a click (success)
k_values = np.arange(0, n + 1)
pmf = binom.pmf(k_values, n, p)
plt.bar(k_values, pmf)
plt.xlabel('Number of Clicks')
plt.ylabel('Probability')
plt.title('Binomial Distribution - Click-Through Rates')
plt.show()

Example 5: Quality Control in Manufacturing

Consider examining 500 light bulbs, of which 5% traditionally exhibit flaws. You need to estimate the likelihood of discovering a specific number of faulty bulbs in the batch.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom
n = 500  # Number of light bulbs
p = 0.05  # Probability of a defective bulb (success)
k_values = np.arange(0, n + 1)
pmf = binom.pmf(k_values, n, p)
plt.bar(k_values, pmf)
plt.xlabel('Number of Defective Bulbs')
plt.ylabel('Probability')
plt.title('Binomial Distribution - Quality Control')
plt.show()