Conditional Probability

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.