Probability


  1. Probability =
       No. of Favourable Outcomes
    Total no. of Possible Outcomes


  2. Random Experiments: Experimental activities, where the result may not be same, when they are repeated under identical conditions. For example, tossing two coins, throwing dice and so on. It has more than one possible outcomes (or results), and it is not possible to predict the outcome in advance

  3. Sample Space: Set of all possible outcomes of a random experiment. In case of throwing a dice,
        S = {1, 2, 3, 4, 5, 6}

  4. Event: Any subset E of a sample space S is called an event. For Example, in case of throwing a dice, if E denotes event of appearance of an odd number then,
        E = {1, 3, 5}

    Types of Events:
  5. Complementary Event: Every Event E has a complementary event E' which includes all elements of sample space, excluding those in E.
        i.e.    E'  =  S - E  =  {ω: ω ∈ S and ω ∉ E}

  6. Event A or B = A ∪ B = {ω :ω ∈ A or ω ∈ B}

  7. Event A and B = A ∩ B = {ω :ω ∈ A and ω ∈ B}

  8. Event A but not B = A - B = A ∩ B' = {ω :ω ∈ A and ω ∉ B}

  9. Mutually Exclusive Events: Two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously.

  10. Exhaustive Events: If A ∪ B ∪ C = S then, A, B, and C are called exhaustive events.

  11. Equally likely outcomes: All outcomes with equal probability

  12. The probability P is a real valued function whose domain is the power set of S and range is the interval [0,1] satisfying the following rules:

  13. P(∅) = 0

  14. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

  15. P[A ∪ B ∪ C] = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)

  16. P(A') = P(not A) = 1 � P(A)

  17. P(A' ∩ B') = P(A ∪ B)' = 1 � P(A ∪ B)


  18. Conditional Probability: Probability of the event E given that F has already occurred. It is denoted by P(E|F).
          P(E|F) =
       Numberof elementaryeventsfavourableto E ∩ F
    Number of elementary events which are favourable to F
      =
       P(E ∩ F)
    P(F)
    ,   provided P(F) ≠ 0



  19. Multiplication rule of probability:
          P(E ∩ F)  =  P(E) . P(F|E)  =  P(F) . P(E|F),
          provided P(E) ≠ 0 and P(F) ≠ 0.

  20. Multiplication rule of probability for more than two events:
          P(E ∩ F ∩ G) ∩= ∩P(E) . P(F|E) . P(G|(E ∩ F))

  21. Independent Events: Two events are independent events if probability of occurrence of one of them is not affected by occurrence of the other.

  22. Theorem of total probability: Let {E1, E2, ....., En} be a partition of a sample space and subpose that each of E1, E2, ....., En has nonzero probability. Let A be any event associated with S, then
         P(A) = P(E1) . P(A|E1) + P(E2) . P(A|E2) + ..... + P(En) . P(A|En)

  23. Bayes' theorem: If E1, E2, ....., En are events which constitute a partition of sample space S, i.e. E1, E2, ....., En are pairwise disjoint and E1 ∪ E2 ∪ ..... ∪ En = S and A is any event of nonzero probability, then,
         P(Ei|A)  =
       P(Ei)P(A|Ei)
    Σ P(Ej)P(A|Ej)
    ,      for any i= 1, 2, 3, ..., n