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course content

Course Content

Probability Theory Basics

1. Basic Concepts of Probability Theory

Stochastic Experiment and Random EventProbability and It's PropertiesGeometrical ProbabilityChallenge: Solving the Task Using Geometric ProbabilityIndependence and Incompatibility of Random EventsConditional Probability

2. Probability of Complex Events

Inclusion-Exclusion PrincipleChallenge: Solving the Task Using Inclusion-Exclusion PrincipleThe Multiplication Rule of ProbabilityLaw of Total ProbabilityBayes' TheoremChallenge: Solving the Task Using Bayes' Theorem

3. Commonly Used Discrete Distributions

Binomial DistributionChallenge: Solving Task Using Binomial DistributionMultinomial DistributionGeometric DistributionPoisson DistributionChallenge: Solving Task Using Poisson Distribution

4. Commonly Used Continuous Distributions

Continuous Uniform DistributionExponential DistributionChallenge: Solving Task Using Exponential DistributionGaussian DistributionChallenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

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

Probability Theory Basics

Independence and Incompatibility of Random EventsIndependence and Incompatibility of Random Events

In probability theory, independence and incompatibility are concepts related to the relationship between random events.

  1. Independence: Two events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the occurrence or non-occurrence of the other event. In other words, knowing whether one event happens provides no information about the likelihood of the other event happening.
    Events A and B are independent if P(A intersection B) = P(A)*P(B);
  2. Incompatibility: Two events are incompatible if they cannot occur simultaneously. If the occurrence of one event excludes the possibility of the other event happening, they are considered incompatible. For example, flipping a coin and getting heads and tails simultaneously is incompatible since the coin can only show one side at a time.
    Events A and B are incompatible if P(A intersection B) = 0.

Examples of independent and incompatible events:

Event Types

Event Types

Independent events Incompatible events
Results of flipping a fair coin and rolling a fair die Rolling a die and getting an odd number and an even number on the same roll
Results of flipping two different coins Selecting a card from a deck and getting a red card and a black card at the same time

You draw a card from a standard deck with replacement (after we have drawn a card, we return it back to the deck) . What is the probability of drawing a red card (heart or diamond) followed by drawing a black card (spade or club)?

Select the correct answer

Everything was clear?

Section 1. Chapter 5
course content

Course Content

Probability Theory Basics

1. Basic Concepts of Probability Theory

Stochastic Experiment and Random EventProbability and It's PropertiesGeometrical ProbabilityChallenge: Solving the Task Using Geometric ProbabilityIndependence and Incompatibility of Random EventsConditional Probability

2. Probability of Complex Events

Inclusion-Exclusion PrincipleChallenge: Solving the Task Using Inclusion-Exclusion PrincipleThe Multiplication Rule of ProbabilityLaw of Total ProbabilityBayes' TheoremChallenge: Solving the Task Using Bayes' Theorem

3. Commonly Used Discrete Distributions

Binomial DistributionChallenge: Solving Task Using Binomial DistributionMultinomial DistributionGeometric DistributionPoisson DistributionChallenge: Solving Task Using Poisson Distribution

4. Commonly Used Continuous Distributions

Continuous Uniform DistributionExponential DistributionChallenge: Solving Task Using Exponential DistributionGaussian DistributionChallenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

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

Probability Theory Basics

Independence and Incompatibility of Random EventsIndependence and Incompatibility of Random Events

In probability theory, independence and incompatibility are concepts related to the relationship between random events.

  1. Independence: Two events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the occurrence or non-occurrence of the other event. In other words, knowing whether one event happens provides no information about the likelihood of the other event happening.
    Events A and B are independent if P(A intersection B) = P(A)*P(B);
  2. Incompatibility: Two events are incompatible if they cannot occur simultaneously. If the occurrence of one event excludes the possibility of the other event happening, they are considered incompatible. For example, flipping a coin and getting heads and tails simultaneously is incompatible since the coin can only show one side at a time.
    Events A and B are incompatible if P(A intersection B) = 0.

Examples of independent and incompatible events:

Event Types

Event Types

Independent events Incompatible events
Results of flipping a fair coin and rolling a fair die Rolling a die and getting an odd number and an even number on the same roll
Results of flipping two different coins Selecting a card from a deck and getting a red card and a black card at the same time

You draw a card from a standard deck with replacement (after we have drawn a card, we return it back to the deck) . What is the probability of drawing a red card (heart or diamond) followed by drawing a black card (spade or club)?

Select the correct answer

Everything was clear?

Section 1. Chapter 5
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