Course Content

Probability Theory Basics

## Probability Theory Basics

# Multinomial Distribution

**The multinomial scheme** extends the Bernoulli trial in cases with more than two outcomes. A multinomial scheme refers to a situation where you have multiple categories or outcomes and are interested in studying the probabilities of each outcome occurring. A probability distribution that models the number of successes in a fixed number of independent trials with multiple categories is called **multinomial distribution**.

## Example

A company is surveying to gather feedback from its customers.

The survey has three possible responses: "Satisfied," "Neutral," and "Dissatisfied." The company randomly selects `50`

customers and records their responses.

Assume that each customer is satisfied with a probability `0.3`

, neutral with a probability `0.4`

, and dissatisfied with a probability `0.3`

.

Calculate the probability that there will be `25`

"Satisfied" responses, `15`

"Neutral," and `10`

"Dissatisfied".

To solve this task multinomial distribution is used:

`import numpy as np from scipy.stats import multinomial # Define the probabilities of each response category probabilities = [0.3, 0.4, 0.3] # Satisfied, Neutral, Dissatisfied # Specify the number of responses for which we calculate probability response = [25, 15, 10] # 25 satisfied, 15 neutral, 10 dissatisfied responses out of 50 total responses # Calculate the probability mass function (pmf) using multinomial distribution pmf = multinomial.pmf(response, n=50, p=probabilities) print(f'Probability of {response}: {pmf:.4f}')`

In the code above, we used `.pmf()`

method of `scipy.stats.multinomial`

class with parameters `n`

(number of trials) and `p`

(probabilities of each outcome) to calculate probability that we will have certain `response`

(the first argument of the `.pmf()`

method.

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