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What is the formula for conditional probability of event A given event B?
What is the formula for conditional probability of event A given event B?
What is the name of the theorem that states P(A|B) = P(B|A) * P(A) / P(B)?
What is the name of the theorem that states P(A|B) = P(B|A) * P(A) / P(B)?
Two events A and B are conditionally independent given event C if?
Two events A and B are conditionally independent given event C if?
What is an application of conditional probability in insurance?
What is an application of conditional probability in insurance?
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What is the probability of both events A and B occurring, given that event B has occurred?
What is the probability of both events A and B occurring, given that event B has occurred?
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What is the name of the rule that states P(A ∩ B) = P(A|B) * P(B)?
What is the name of the rule that states P(A ∩ B) = P(A|B) * P(B)?
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Study Notes
Conditional Probability
Definition
Conditional probability is the probability of an event occurring given that another event has occurred.
Formula
P(A|B) = P(A ∩ B) / P(B)
- P(A|B) is the probability of event A occurring given that event B has occurred
- P(A ∩ B) is the probability of both events A and B occurring
- P(B) is the probability of event B occurring
Properties
- Chain Rule: P(A ∩ B) = P(A|B) * P(B)
- Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B)
Types of Conditional Probability
- Conditional Independence: Two events A and B are conditionally independent given event C if P(A ∩ B|C) = P(A|C) * P(B|C)
- Conditional Dependence: Two events A and B are conditionally dependent given event C if P(A ∩ B|C) ≠ P(A|C) * P(B|C)
Real-World Applications
- Medical diagnosis: determining the probability of a disease given a positive test result
- Insurance: calculating the probability of an accident given a driver's age and experience
- Marketing: predicting the probability of a customer purchasing a product given their demographics and behavior
Conditional Probability
- Conditional probability is the probability of an event occurring given that another event has occurred.
Formula
- The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B), where P(A|B) is the probability of event A occurring given that event B has occurred, P(A ∩ B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.
Properties
- The Chain Rule states that P(A ∩ B) = P(A|B) * P(B), which is a useful formula for calculating the probability of both events occurring.
- Bayes' Theorem states that P(A|B) = P(B|A) * P(A) / P(B), which allows for the calculation of the probability of an event A given event B, given the probability of event B given event A, and the individual probabilities of events A and B.
Types of Conditional Probability
- Two events A and B are conditionally independent given event C if P(A ∩ B|C) = P(A|C) * P(B|C), meaning that the occurrence of event C does not affect the probability of events A and B occurring together.
- Two events A and B are conditionally dependent given event C if P(A ∩ B|C) ≠ P(A|C) * P(B|C), meaning that the occurrence of event C affects the probability of events A and B occurring together.
Real-World Applications
- Conditional probability is used in medical diagnosis to determine the probability of a disease given a positive test result.
- It is used in insurance to calculate the probability of an accident given a driver's age and experience.
- It is used in marketing to predict the probability of a customer purchasing a product given their demographics and behavior.
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Description
Understand the concept of conditional probability, its formula, and properties such as the chain rule and Bayes' theorem.
