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Random Experiment, Sample Space and Events
  Probability is the likelihood or chance that something is the case or will happen.
Random Experiment : An activity or process whose value cannot be predicted ahead of time.

Sample space: The set of all possible outcomes of a given random experiment is called the sample space associated with that experiment. Each possible outcome or element in a sample space is called a sample point or an elementary event.

Event :

Every non-empty subset A of S, which is a disjoint union of single element subsets of the sample space S of a random experiment E is called an event.
Probability of an outcome an event =

Probability Function :

P(A) is the probability function defined on a σ field B of events if the following properties hold:
  • For each A ε B, P(A) is defined as real and P(A) ≥ 0.
  • P(S) = 1
  • If {An} is any finite or infinite sequence of disjoint events in B, then

    Algebra of Events:

    for events A, B, C
    1. A B = {w ε S: w ε A or w ε B}
    2. A B = { w ε S: w ε A and w ε B}
    3. = {w ε S: w A}
    4. A – B = {w ε S: w ε A but w B}
    5. A B for every w ε A, w ε B
    6. B A A B
    7. A = B if and only if A and b have same elements, i.e., A B and B A.
    8. A and B disjoint (mutually exclusive) A B = (null set)
    9. A B can be denoted as A + B if A and B are disjoint.
    10. A Δ B denotes those w belonging to exactly one of A and B, i.e.,
    11. A Δ B = B A = B + A (disjoint events)

Some Theorems on Probability :

  1. Probability of the impossible event (null event) is zero, i.e., P() = 0.
  2. Probability of the complementary event of A is given by P( ) = 1 – P(A)
  3. For any two events A and B, we have
    a. P( B) = P(B) – P(A B)
    b. P(A ) = P(A) – P(A B)

  4. If B A, then
    a) P(A ) = P(A) – P(B)
    b) P(B) ≤ P(A)

  5. Addition Theorem of Probability: If A and B are any two disjoint events (subsets of sample space S), then P(A B) = P(A) + P(B) – P(A B)
  6. Boole’s Inequality: For n events A1, A2, …..,An, we have


  7. Multiplication Theorem of Probability: If A and B are two events, the probability of their joint occurrence, i.e.,
    P(A and B), is: P(A and B) = P(A) P(B | A)

  8. Bayes’ Theorem: Bayes' theorem is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. The probability of an event A conditional on another event B is generally different from the probability of B conditional on A. However, there is a definite relationship between the two, and Bayes' theorem is the statement of that relationship.
  
 

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