In this lesson we will look at some laws or formulas of probability: the Addition Law, the Multiplication Law and the Bayes’ Theorem or Bayes’ Rule.
The general law of addition is used to find the probability of the union of two events. The expression denotes the probability of X occurring or Y occurring or both X and Y occurring.
The Addition Law of Probability is given by
where X and Y are events.
If the two events are mutually exclusive, the probability of the union of the two events is the probability of the first event plus the probability of the second event. Since mutually exclusive events do not intersect, nothing has to be subtracted.
If X and Y are mutually exclusive, then the addition law of probability is given by
The probability of the intersection of two events is called joint probability.
The Multiplication Law of Probability is given by
The notation is the intersection of two events and it means that both X and Y must happen. denotes the probability of X occurring given that Y has occurred.
When two events X and Y are independent,
If X and Y are independent then the multiplication law of probability is given by
The Bayes’ Theorem was developed and named for Thomas Bayes (1702 – 1761). Bayes’ rule enables the statistician to make new and different applications using conditional probabilities. In particular, statisticians use Bayes’ rule to ‘revise’ probabilities in light of new information.
The Bayes’ theorem is given by
Bayes’ theorem can be derived from the multiplication law
Bayes’ Theorem can also be written in different forms
The following video gives an intuitive idea of the Bayes' Theorem formulas: we adjust our perspective (the probability set) given new, relevant information. Formally, Bayes' Theorem helps us move from an unconditional probability to a conditional probability.