Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. A Bernoulli trial is an experiment with only two possible outcomes, such as heads or tails, success or failure, yes or no, etc.

A binomial experiment has the following properties:

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• The experiment consists of n independent trials, each with the same probability of success, denoted by p.

• The outcome of each trial is either success (denoted by 1) or failure (denoted by 0).

• The number of successes in the n trials is of interest.

 

The binomial distribution is defined by two parameters: n, the number of trials, and p, the probability of success. The probability mass function (PMF) of the binomial distribution is given by:

                                               

                                                P(X = k) = (n choose k) * p^k * (1 - p)^(n-k), for k = 0, 1, 2, ..., n

 

                   where (n choose k) is the binomial coefficient, which is given by n! / (k! * (n-k)!).

                   The mean (expected value) of the binomial distribution is given by:

 

                                                                                  E(X) = np

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                   The variance of the binomial distribution is given by:

 

                                                                                Var(X) = np * (1 - p)

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The binomial distribution can be used to model a wide range of real-world experiments, including the number of heads in a sequence of coin tosses, the number of customers who make a purchase in a given time period, and the number of defective items in a batch of products.

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here are a few examples of situations that could be modeled using a binomial distribution:

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1. The number of heads in 10 tosses of a fair coin.

2. The number of customers who make a purchase after clicking on an advertisement.

3. The number of defective items in a sample of 100 products.

4. The number of patients who experience a side effect after taking a certain medication.

 

In each of these examples, the outcome can only be one of two possibilities (e.g. heads or tails, purchase or no purchase, defective or not defective, etc.), and the number of trials is fixed. These properties make the binomial distribution a good fit for modeling the data.

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Here is an implementation of the binomial distribution in Python:

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import math

def binomial_distribution(n, p, k):
  return (math.comb(n, k) * (p ** k) * ((1 - p) ** (n - k)))

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Where n is the number of trials, p is the probability of success in each trial, and k is the number of successful outcomes you're interested in finding the probability for. The math.comb function computes the number of combinations of n things taken k at a time.

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For example, to find the probability of getting exactly 5 heads in 10 tosses of a fair coin:

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n = 10
p = 0.5
k = 5

prob = binomial_distribution(n, p, k)
print("Probability:", prob)
 

This will return Probability: 0.24609375.