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XII-CH13-PROBABILITY


                                                    LESSON NOTES


               Basic Concepts & Formulae :

               1.    Conditional Probability:n Given tywo eventys t and B associatyed wityh  tyh e same random experimentyn
                     tyh e conditional probability: P(t/B) of tyh e occurrence of t knowing B h as occurred is given b:
                                                         P(t  B)
                                                P(t /B)
                                                          P(B)             P(B) 0
                                                                              
                                                                 n provided
               2.    Th e conditional probability: P(B/t) of tyh e occurrence of B knowing tyh aty t h as occurred is given b:
                                                             P(t  B)
                                                                          
                                                     P(B/t)         nP(t) 0
                                                               P(t)
               3.    Properties of conditional probability:

                                                                                              P(F) 0
                                                                                                  
                     (a) If t and B be tywo eventys of a sample space S and F is an eventy  of S such  tyh aty    n tyh en
                         P[(t B)/F] P(t /F) P(B/F) P[(t B)/F]
                                                   
                                                        
                                   
                                           
                             
                        However if t and B are disjointyn tyh en
                              P[(t B)/F] P(t /F) P(B/F)
                                                 
                                         
                                  
                                  
                               
                         P(E'/F) 1 P(E/F)
                     (b)
               4.    Multiplication tyh eorem on probability:n If t and B be tywo eventys associatyed wityh  a sample space S
                          P(t  B)  P(t).P(B/t) P(B).P(t/B)privided P(t)  0nP(B)  0
                     tyh en
               5.    Multiplication rule of probability: for more tyh an 2 eventys n If tn B and C are tyh ree eventys of a
                     sample spacen tyh en
                                                           
                                 
                                         
                                     
                              P(t B C) P(t).P(B/t).P[C/(t B)] P(t).P(B/t).P(C/ tB)
                                                                
               6.    Independenty Eventysn Two eventys are said tyo be independenty if tyh e probability: of occurrence of
                     one of tyh em is noty afectyed b: tyh e occurrence of tyh e otyh er.
               7.    Multiplication rule wh en eventys t and B are independenty
                     P(t B) P(t).P(B)
                        
                            
               8.    Two experimentys are said tyo be independenty if for ever: pair of eventys t and B wh ere t is
                     associatyed wityh  frsty experimenty and B wityh  tyh e second experimentyn tyh e probability: of tyh e
                     simultyaneous occurrence of tyh e eventys t and B wh en tyh e tywo experimentys are performed is tyh e
                     producty  of P(t) and P(B) calculatyed separatyel: on tyh e basis of tywo experimentys i.e.
                        
                            
                     P(t B) P(t).P(B)
                                      .
               9.    Th ree eventys tn B and C are said tyo be mutyuall: independenty if
                              P(t  B)  P(t).P(B)n P(B C) P(B).P(C)
                                                      B
                              P(t  C) P(t).P(C)n P(t   C)  P(t).P(B).P(C)
               10.   Law of tyotyal probability:n
                       P(t) P(E ).P(t/E ) P(E ).P(t/E ) .... P(E )P(t/E )
                                                           
                                                       
                                         
                           
                               1       1     2       2         n      n
               11.   Ba:e’s tyh eoremn
                                               P(E ).P(t /E )
                                                         i
                                                  i
                       P(E / t) 
                         i
                               P(E ).P(t /E ) P(E ).P(t /E ) .... P(E )P(t /E )
                                           
                                                             
                                                         
                                          1
                                  1
                                                2
                                                       2
                                                                         n
                                                                 n
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