Conditional Probability

Fundamentals of Conditional Probability

  • Conditional probability is the calculation of the probability of an event given the occurrence of another event.
  • It is denoted by **P(A B)**, which is read as ‘the probability of A given B’.
  • The formula for calculating conditional probability is **P(A B) = P(A ∩ B) / P(B)**. This formula assumes that P(B) isn’t zero, to avoid division by zero.

Independent and Dependent Events

  • Events A and B are said to be independent if the occurrence of A does not affect the occurrence of B, and vice versa.
  • In such cases, the conditional probability P(A B) is simply P(A), and the conditional probability P(B A) is P(B).
  • If events are not independent, they are said to be dependent. The occurrence of one does affect the probability of the occurrence of the other.

Reversed Conditional Probability

  • If you are asked to find the probability of event A given event B when you know the probability of event B given A, you can use Bayes’ Theorem.
  • Bayes’ Theorem states that **P(A B) = [P(A) × P(B A)] / P(B)**.

Relationships Between Events and Conditional Probabilities

  • If two events are mutually exclusive, then the conditional probability of one event occurring given the other has occurred is 0.
  • If two events are exhaustive, meaning that at least one of them must occur, then P(A B) = 1 - P(A’), where A’ is the complement of A (the event that A does not occur).

Practical Applications of Conditional Probability

  • Conditional probability is frequently used in fields such as medicine, psychology, weather forecasting, and gambling.
  • Always make sure you understand the given circumstances when working with conditional probabilities. Misunderstanding the context can lead to incorrect results.

Remember, the mastery of conditional probability comes with meticulous practice. Continually tackle a wide range of practical problems until you gain confidence.