Conditional Probability

Defining Conditional Probability

  • Conditional Probability measures the probability of an event occurring, considering that another event has already occurred.
  • It is expressed as **P(A B)**, which reads as the probability of event A given that event B has occurred.
  • In this notation, event A is the event we are interested in, and event B is the event that has already occurred.

Calculating Conditional Probability

  • The formula to calculate conditional probability is: **P(A B) = P(A ∩ B) / P(B)**
  • In this formula, ∩ represents intersection, meaning both events A and B occur.
  • This formula assumes that P(B) is not equal to 0, as division by zero is undefined.
  • P(A ∩ B) signifies the probability of both events A and B occurring.

Independent and Dependent Events

  • If events A and B are independent, knowing that B has occurred does not affect the probability of A. Therefore, P(A B) = P(A).
  • If events A and B are dependent, knowing that B has occurred changes the probability of A. We need to use the formula for conditional probability to find P(A B).

Using Tree Diagrams to Solve Conditional Probability Problems

  • Tree diagrams can help visualise all possible outcomes of two or more events and can be particularly useful in solving conditional probability problems.
  • Each branch of the tree represents a possible outcome.
  • The probability of each branch is written on the branch.
  • The end of the branches represent final outcomes and the probabilities are calculated by multiplying the probabilities along the branches.

The Multiplication Rule for Conditional Probability

  • The multiplication rule states that the probability of both of two dependent events happening, A and B, is the probability of A times the probability of B given A.
  • The formula is **P(A ∩ B) = P(A) ・ P(B A)**.
  • Conversely, it can also be expressed as **P(A ∩ B) = P(B) ・ P(A B)**.

Bayes’ Theorem

  • Bayes’ Theorem is an important concept in probability theory and statistics that describes how to update the probabilities of hypotheses when given new data.
  • Formally, the theorem is defined as **P(A B) = [P(B A) ・ P(A)] / P(B)**.
  • It gives a way to revise existing predictions or theories (posterior probability) given new or additional evidence.