Probability Theory Basics

Challenge: Solving the Task Using Bayes' Theorem

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Imagine that we decided to conduct medical research.
You gathered data about two groups: 750 people with heart problems and 800 with chronic stomachache.
You know that 7% of interviewed from the first group have diabetes; meanwhile, 12% of respondents from the second group have diabetes too.
Calculate the probability that a randomly selected person with diabetes has a chronic stomachache.

We can use Bayes' theorem to calculate the corresponding probability.

Calculate the probability that a randomly selected person has a heart problem.
Calculate the probability that the randomly selected person has a stomachache.
Calculate the probability that you randomly select a person that has diabetes.

Soluzione

# Probability that ramdomly selected person has heart problem

P_heart = 750 / 1550

# Probability that ramdomly selected person has stomachache

P_stomach = 800 / 1550

P_heart_diabetes = 0.07

P_stomach_diabetes = 0.12

# Probability that you randomly select a person that has diabetes

P_random_diabetes = P_heart * P_heart_diabetes + P_stomach * P_stomach_diabetes

# Randomly selected person with diabetes has chronic stomachache

P_diabetes_belong_stomach = (P_stomach * P_stomach_diabetes)/P_random_diabetes

print(f'The probability is {P_diabetes_belong_stomach:.4f}')

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Sezione 2. Capitolo 6

# Probability that randomly selected person has heart problem

P_heart = ___ / 1550

# Probability that randomly selected person has stomachache

P_stomach = ___

P_heart_diabetes = 0.07

P_stomach_diabetes = 0.12

# Probability that you randomly select a person that has diabetes

P_random_diabetes = ___ * P_heart_diabetes + P_stomach * ___

# Randomly selected person with diabetes has chronic stomachache

P_diabetes_belong_stomach = (P_stomach * P_stomach_diabetes)/P_random_diabetes

print(f'The probability is {P_diabetes_belong_stomach:.4f}')

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Probability Theory Basics

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1. Basic Concepts of Probability Theory

We will start our way of learning probability theory by considering some basic definitions and rules: what is a stochastic experiment and random event, what is independence and incompatibility of events in the context of probability theory, what is the probability and how can we calculate probabilities of different elementary events.

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

In real-life tasks, we often have to deal with complex relationships and, as a result, calculate probabilities of several events or events that depend on each other. Let's consider how we can do this using probability theory.

Inclusion-Exclusion Principle

Challenge: Solving the Task Using Inclusion-Exclusion Principle

The Multiplication Rule of Probability

Law of Total Probability

Challenge: Solving the Task Using Bayes' Theorem

3. Commonly Used Discrete Distributions

To solve many real problems in probability theory, special models have been created that describe a particular situation. Let's consider some of the most used models that can be used to describe some discrete results of stochastic experiments.

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

What if the result of a stochastic experiment cannot be described by a discrete value? For this, models that work with continuous values are used. Consider the most popular of these models.

Continuous Uniform Distribution

Exponential Distribution

Challenge: Solving Task Using Exponential Distribution

Gaussian Distribution

Challenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

Often we are faced with the task of checking the dependence of the results of different stochastic experiments on each other. Moreover, it is necessary not only to assess the presence of dependencies but also to somehow quantify the degree of dependencies. To solve these problems, we can use covariance and correlation.

What is Covariance?

What is Correlation?

Challenge: Solving the Task Using Correlation