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# Probability is a chance of occurrence

Probability is a chance of occurrence
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Probability is a chance of occurrence . Sample space versus an event (they are just like a population versus a sample) A sample space (S) is a set of all possible events or all possible outcomes of arandom experiment. An event is each possible outcome of a variable.(mostly denoted by capital capital letters, say A or B or C etc.Example 1. When you roll a die as an experiment, all possible outcomes are S ={ 1,2,3,4,5,6 } Let event  A  = { 1,2 } and  B ={ 2,4 6 } Assigning probability to an event.  Assigning probability to event A. Using the previous example (example 1) S ={ 1,2,3,4,5,6 }  the possibility or chance of getting event A is { 1,2 }  out of  { 1,2,3,4,5,6 } .  Number of elements in A is 2, n (  A )= 2  and number of elementsin S is 6, n ( S )= 6  so chance of event A occurring is given by:  P (  A  ) = n (  A  ) n ( S ) = 26 = 13 = 0.33 Do the same for all events. Assigning probability to event B. S ={ 1,2 , 3,4 , 5,6 }  the possibility or chance of getting event B is { 2,4,6 }  out of  { 1,2,3,4,5,6 } .  Number of elements in B is 3, n ( B )= 3  and number of elementsis S is 6, n ( S )= 6  so chance of event A occurring is given by:   P ( B ) = n ( B ) n ( S ) = 36 = 12 = 0.5 The probability of a sample space is always equal to one,   P ( S ) = n ( S ) n ( S )= 66 = 1  Rules of probabilities .Let A and B be any events. Complement can be denoted by  A  c or ´  A   or  A  '  “or” means union ∪  (A or B means all elements in A and B)“and” means intersection ∩  (A and B common elements in A and B)1.Complementary:  P (  A  c ) = 1 −  P (  A  )  or  P ( B c ) = 1 −  P ( B ) 2.Additional:  P (  A ∪ B )=  P (  A ) +  P ( B )−  P (  A ∩B )  3.Conditional:  P (  A  / B ) =  P (  A∩B )  P ( B )  or ( B /  A  ) =  P (  A∩B )  P (  A  )  .Note that  P (  A∩B ) =  P ( B∩A  )  and  P (  A  ∪ B ) =  P ( B ∪  A  ) Conditions of probabilities 1.Mutually exclusive : there is no intersection between the events, that is  P (  A∩B ) = 0 2.Independent : one event does not depend on the other to occur, that is  P (  A∩B ) =  P (  A ) ×P ( B ) Let A and B be mutually exclusive events. 1. Complementary:    P (  A  c ) = 1 −  P (  A  )  or  P ( B c ) = 1 −  P ( B )   Is not affected by the condition because it does not contain  P (  A∩B ) . 2.  Additional  :  P (  A  ∪ B ) =  P (  A  ) +  P ( B ) −  P (  A∩B ) =  P (  A  ) +  P ( B ) − 0 ¿  P (  A ) +  P ( B )   3. Conditional:  P (  A  / B ) =  P (  A∩B )  P ( B ) =  0  P ( B )= 0   or  P ( B /  A  ) =  P (  A∩B )  P (  A  ) =  0  P (  A  )= 0  .  Let A and B be independent events.1.Complementary:  P (  A  c ) = 1 −  P (  A  )   or  P ( B c ) = 1 −  P ( B )   Is not affected by the condition because it does not contain  P (  A∩B ) . 2.Additional  :  P (  A  ∪ B ) =  P (  A  ) +  P ( B ) −  P (  A∩B ) =¿    P (  A ) +  P ( B ) −[  P (  A ) ×P ( B ) ]  3. Conditional:    P (  A  / B ) =  P (  A∩B )  P ( B ) =  P (  A  ) ×P ( B )  P ( B ) =  P (  A  ) or     P ( B /  A  ) =  P (  A∩B )  P (  A  ) =  P (  A  ) ×P ( B )  P (  A  ) =  P ( B )  . Contingency table Events  B  ´ B total  A   P (  A∩B )  P (  A∩ ´ B )  P (  A  ) ´  A   P ( ´  A∩B )  B ´  A∩ ´¿  P ¿  P ( ´  A  ) total  P ( B )  P ( ´ B )  1
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