1. Random Experiments and Sample Space
Random Experiment: An experiment whose outcome cannot be predicted with certainty (e.g., rolling a die, tossing a coin).
Sample Space (S): The set of all possible outcomes of a random experiment. Each outcome is a sample point.
2. Events
An event is any subset of the sample space.
- Impossible Event: The empty set ∅. Probability = 0.
- Sure Event: The entire sample space S. Probability = 1.
- Mutually Exclusive Events: Events A and B cannot occur simultaneously. A ∩ B = ∅.
- Exhaustive Events: Events A₁, A₂... are exhaustive if A₁ ∪ A₂ ∪ ... = S.
3. Axiomatic Approach to Probability
Probability P is a real-valued function whose domain is the power set of S, satisfying:
- 0 ≤ P(E) ≤ 1 for any event E.
- P(S) = 1.
- If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).
4. Addition Theorem of Probability
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
For any two events A and B.
Probability of 'Not A'
The probability of the complement of A (denoted A') is given by:
P(A') = 1 - P(A)