Bayesian Probability & Inference

Master conditional probability, total probability, and Bayes' theorem. Learn to update beliefs with evidence and apply Bayesian reasoning to real-world problems including medical diagnosis and classification.

12th Grade

Statistics

~90 min

🎮 Interactive Activity: Bayes' Theorem Calculator

Calculate the posterior probability using Bayes' theorem!

Given:

P(Disease) = 0.01

P(Test+|Disease) = 0.95

P(Test-|No Disease) = 0.95

Find: P(Disease | Test+)

P(Disease | Test+) = ?

🎮 Interactive Activity: Conditional Probability Finder

Find the conditional probability!

Given:

P(A) = 0.30

P(B|A) = 0.80

P(B|¬A) = 0.40

Find: P(A|B)

P(A|B) = ?

1. Conditional Probability

Understanding Conditional Probability

Conditional probability P(A|B) measures the probability of event A occurring given that event B has occurred. It's fundamental to Bayesian inference and updating beliefs with new information.

Example 1: Basic Conditional Probability

Given: P(Rain) = 0.3, P(Clouds|Rain) = 0.9, P(Clouds|No Rain) = 0.4

Find: P(Rain|Clouds)

Step 1: P(Clouds) = 0.9×0.3 + 0.4×0.7 = 0.55

Step 2: P(Rain|Clouds) = (0.9×0.3)/0.55 ≈ 0.491

Answer: 0.491

Example 2: Card Drawing

Given: Draw a card from a standard deck

Find: P(King|Face Card)

Step 1: P(Face Card) = 12/52 = 3/13

Step 2: P(King and Face Card) = 4/52 = 1/13

Step 3: P(King|Face Card) = (1/13)/(3/13) = 1/3

Answer: 1/3

Example 3: Independence vs Dependence

Independent events: P(A|B) = P(A) - knowing B doesn't change A's probability

Dependent events: P(A|B) ≠ P(A) - knowing B changes A's probability

Example: Drawing two cards without replacement: second card depends on first

2. Law of Total Probability

3. Bayes' Theorem

4. Interpreting Prior and Posterior

5. Naive Bayes Classification

6. Real-World Applications

7. Advanced Bayesian Concepts

Frequently Asked Questions

Practice Time!

Practice Quiz

Questions

Correct

Score

What is Bayes' theorem formula?

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If P(Disease) = 0.01, P(Test+|Disease) = 0.95, and P(Test-|No Disease) = 0.95, what is P(Test+)?

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In the medical test example above, what is P(Disease|Test+)?

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What is the Law of Total Probability?

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What is conditional probability P(A|B)?

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If P(A) = 0.3, P(B|A) = 0.8, and P(B|¬A) = 0.4, what is P(A|B)?

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What does 'prior probability' mean in Bayesian inference?

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What does 'posterior probability' mean?

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In Naive Bayes classification, what assumption is made?

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If three events A₁, A₂, A₃ partition the sample space with P(A₁)=0.5, P(A₂)=0.3, P(A₃)=0.2, and P(B|A₁)=0.2, P(B|A₂)=0.5, P(B|A₃)=0.7, what is P(B)?

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