Introduction to Probability

Definition:

• Probability is a measure of the likelihood that an event will occur.
• It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Sample Space and Events:

• The sample space, denoted by S, is the set of all possible outcomes of an experiment.
• An event is any subset of the sample space.

Probability of an Event:

• The probability of an event A, denoted by P(A), is the sum of the probabilities of all outcomes in A.
• It satisfies the following properties:
  1. 0 ≤ P(A) ≤ 1 for any event A.
  2. P(S) = 1, where S is the sample space.
  3. If A and B are disjoint events (i.e., they have no outcomes in common), then P(A ∪ B) = P(A) + P(B).

Probability Rules:

• Complement Rule: The probability of the complement of an event A, denoted by A' or A^c, is P(A') = 1 - P(A).
• Union Rule: The probability of the union of two events A and B, denoted by A ∪ B, is P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• Intersection Rule: If A and B are independent events, then P(A ∩ B) = P(A) × P(B).

Types of Probability:

• Classical Probability: Based on equally likely outcomes in a sample space.
• Empirical Probability: Based on observed frequencies from data.
• Subjective Probability: Based on personal judgment or opinion.

Conditional Probability:

• Conditional probability measures the likelihood of an event occurring given that another event has already occurred.
• It is denoted by P(A|B) and calculated as P(A|B) = P(A ∩ B) / P(B).

Independence:

• Two events A and B are independent if the occurrence of one event does not affect the occurrence of the other.
• Mathematically, P(A ∩ B) = P(A) × P(B).