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Probability Theory Basics

1. Basic Concepts of Probability Theory

Stochastic Experiment and Random Event Probability and It's Properties Geometrical Probability Challenge: Solving the Task Using Geometric Probability Independence and Incompatibility of Random Events Conditional Probability

2. Probability of Complex Events

Inclusion-Exclusion Principle Challenge: Solving the Task Using Inclusion-Exclusion Principle The Multiplication Rule of Probability Law of Total Probability Bayes' Theorem Challenge: Solving the Task Using Bayes' Theorem

3. Commonly Used Discrete Distributions

Binomial Distribution Challenge: Solving Task Using Binomial Distribution Multinomial Distribution Geometric Distribution Poisson Distribution Challenge: Solving Task Using Poisson Distribution

4. Commonly Used Continuous Distributions

Continuous Uniform Distribution Exponential Distribution Challenge: Solving Task Using Exponential Distribution Gaussian Distribution Challenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

What is Covariance?What is Correlation?Challenge: Solving the Task Using Correlation

Binomial Distribution

Discrete distribution describes the experiment with a finite number of possible outcomes.

Imagine a stochastic experiment with only two possible outcomes: success and failure.
A successful result for us appears with a probability of p, respectively, and an unsuccessful result appears with a probability of 1-p.
Such an experiment is called Bernoulli trial. A sequence of several independent Bernoulli trials is called a Bernoulli process.
Assume that we are flipping a coin and want to have a tail. Such a flip will be a Bernoulli trial. If we flip the coin 2 or more times, it will be the Bernoulli process.

We can conduct the Bernoulli process in Python using the .rvs() method of the scipy.stats.bernoulli class.
Assume we have an asymmetrical coin in which the probabilities of getting a head and a tail are unequal. We can simulate 10 tosses of such coin using the following code:


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from scipy.stats import bernoulli

# Define the probability of success (getting a head for example)
probability = 0.3

# Generate Bermoulli process
random_samples = bernoulli.rvs(size=10, p=probability)
print(f'Bernoulli process: {random_samples}')

In .rvs() method above we specify size and p parameters. The first parameter is used to specify the number of values to generate and the second is used to specify the probability of success.

The binomial distribution is a discrete probability distribution that describes the number of successes (k) in a Bernoulli process with a fixed number of trials (n).

Let's look at the example:


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from scipy.stats import binom

# Parameters
n_samples = 10  # Number of trials
proba = 0.5  # Probability of success

# Calculate the probability that there are k successes
k = 3  # Number of successes
pmf = binom.pmf(k, n=n_samples, p=proba)
print(f'Probability of getting {k} successes: {pmf:.4f}')

We used .pmf() method of the scipy.stats.binom class with parameters n (number of trials) and p(probability of success) to calculate the probability that we have exactly 3 (the first argument of .pmf() method) successes in 10 Bernoulli trials. We will consider .pmf() method in more detail in Probability Theory Mastering course.

Note
It is important to understand the difference - the Bernoulli trial is a stochastic experiment with certain properties, while the binomial distribution is the rule by which we can calculate the probabilities of occurring events in the Bernoulli process.

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