Probability

1. Probability =

   	No. of Favourable Outcomes
   	Total no. of Possible Outcomes

   
   
   
3. 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
   
   
4. Sample Space: Set of all possible outcomes of a random experiment. In case of throwing a dice,
       S = {1, 2, 3, 4, 5, 6}
   
   
5. 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:
   • Impossible Events: The empty set ∅ denotes impossible event. Like having a 7 on throwing a dice.
     
     
   • Sure Events: The whole sample space, S, is called the sure event.
     
     
   • Simple (or elementary) Event: Event E having only one sample point of a sample space.
     
     
   • Compound Event: Event E with more than one sample point of sample space.
     
     
6. 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}
   
   
7. Event A or B = A ∪ B = {ω :ω ∈ A or ω ∈ B}
   
   
8. Event A and B = A ∩ B = {ω :ω ∈ A and ω ∈ B}
   
   
9. Event A but not B = A - B = A ∩ B^' = {ω :ω ∈ A and ω ∉ B}
   
   
10. 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.
    
    
11. Exhaustive Events: If A ∪ B ∪ C = S then, A, B, and C are called exhaustive events.
    
    
12. Equally likely outcomes: All outcomes with equal probability
    
    
13. 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:
    • For any event E, P (E) = 0
    • P (S) = 1
    • If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F).
    
    
    
14. P(∅) = 0
    
    
15. P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
    
    
16. P[A ∪ B ∪ C] = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)
    
    
17. P(A') = P(not A) = 1 � P(A)
    
    
18. P(A' ∩ B') = P(A ∪ B)' = 1 � P(A ∪ B)
    
    

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19. 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
    
    
    • P(S|F)  =  P(F|F)  =  1
    • P((A ∪ B)|F) = P(A|F) + P(B|F) � P((A ∩ B)|F),   provided P(F) ≠ 0
    • P(E'|F)  =  1  -  P(E|F)
    
    
    
20. 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.
    
    
21. Multiplication rule of probability for more than two events:
          P(E ∩ F ∩ G) ∩= ∩P(E) . P(F|E) . P(G|(E ∩ F))
    
    
22. Independent Events: Two events are independent events if probability of occurrence of one of them is not affected by occurrence of the other.
    
    • Thus, P(F|E)  =  P(F), provided P(E) ≠ 0
      and,   P (E|F)  =  P(E), provided P (F) ≠ 0
    • Thus, P(E ∩ F) = P(E) . P (F|E) = P(E) . P(F)
    • Three events A, B and C are said to be mutually independent, if
                  P(A ∩ B)  =  P(A) . P(B)
                  P(A ∩ C)  =  P(A) . P(C)
                  P(B ∩ C)  =  P(B) . P(C)
           and, P(A ∩ B ∩ C)  =  P(A) . P(B) . P(C)
      
    
    
23. Theorem of total probability: Let {E₁, E₂, ....., E_n} be a partition of a sample space and subpose that each of E₁, E₂, ....., E_n 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(E_n) . P(A|E_n)
    
    
24. Bayes' theorem: If E₁, E₂, ....., E_n are events which constitute a partition of sample space S, i.e. E₁, E₂, ....., E_n are pairwise disjoint and E₁ ∪ E₂ ∪ ..... ∪ E_n = S and A is any event of nonzero probability, then,
         P(E_i|A)  =

    	P(Ei)P(A|Ei)
    	Σ P(Ej)P(A|Ej)

    ,      for any i= 1, 2, 3, ..., n