Probability Fundamentals

For all the definitions below, consider as events of a sample space .

Probability of an event

Suppose is an event of the sample space .

Can be in the range .

Note

If and are mutually exclusive events, then .

Addition Law

Marginal probability

The probability of an event occurring without any additional information or conditions from other events. Useful when dealing with joint probability distributions and when analyzing how events relate to each other.

Conditional probability

The probability of an event occurring, given that another event has already occurred.

Where:

• is the conditional probability of given
• is the joint probability of both and occurring
• is the probability of event occurring

Probability assessments can be updated when new information becomes available through conditional probability. It is particularly useful in scenarios where events are dependent on one another.

Note

If events and are independent, then , meaning the occurrence of doesn’t affect the probability of .

Bayes’ Theorem

Suppose and are two events.

Can only be applied when:

• The sample space is partitioned into a set of mutually exclusive events: .
• is to be calculated
• At least one of the two sets of possibilities should be given:
  • and

Multiplication theorem

Law of total probability

Relates marginal probablities to conditiional probablities.

Suppose the sample space is partitioned into a countably infinite set of mutually exclusive events: . Then, for an event :

Joint Probability

Probability of 2 or more events occuring simultaneously.

Note

For discrete random variables, the marginal probability can be calculated by summing over the joint probability distribution:

For continuous random variables: