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Lecture 5

by: Debra Tee

Lecture 5 STATS 250

Debra Tee
GPA 3.85

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Probability, Probability Definitions and Relationships, Probability Rules and Formulas, Sampling with and without replacement.
Introduction to Statistics
Brenda Gunderson
Class Notes
Probability, Sampling, Rules, formulas
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This 3 page Class Notes was uploaded by Debra Tee on Monday October 3, 2016. The Class Notes belongs to STATS 250 at University of Michigan taught by Brenda Gunderson in Fall 2016. Since its upload, it has received 2 views. For similar materials see Introduction to Statistics in Statistics at University of Michigan.


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Date Created: 10/03/16
Lecture  5:  Probability     7.1  Random  Circumstance  and  Interpretations  of  Probability     -­  A  few  ways  to  think  about  PROBABILITY:   (1)  Personal  or  Subjective  Probability   -­  P(A)  =  the  degree  to  which  a  given  individual  believes  that  the  event  A  will   happen.     (2)  Long  term  relative  frequency   P(A)  =  proportion  of  times  ‘A’  occurs  if  the  random  experiment   (circumstance)  is  repeated  many,  many  times.     (3)  Basket  Model   P(A)  =  proportion  of  balls  in  the  basket  that  have  an  ‘A’  on  them.     -­   Note:  each  time  I  do  the  experiment,  the  selected  ball  is  either  white   or  blue;;  once  I  look,  there  is  no  more  ‘probability’)     -­   Note:  A  probability  statement  IS  NOT  a  statement  about   INDIVIDUALS.  It  IS  a  statement  about  the  population  /  the  basket  of   balls  .         7.3  Probability  Definitions  and  Relationships   7.4  Basic  Rules  for  Finding  Probability     -­  probability  of  any  outcome  is  always  between  0  and  1     Probability  Rules  and  Formulas     -­‐    Complement  rule  P(AC  )    ▯ 1  -­    P(A)   -­‐   Addition  rule  P(A  or  B)  =  P(A)  +  P(B)  -­▯  P(A  and  B)   -­‐   Multiplication  rule  P(A  and  B)  =▯  P(A)P(B  |  A)   -­‐   Conditional  Probability  P(AB)  =  P(A  and  B)/P(B)     Definition:   -­‐   Two  events  A,  B  are  Mutually  Exclusive  (or  Disjoint)  if  they  do  not   contain  any  of  the  same  outcomes.  So  their  intersection  is  empty.   -­‐   Two  events  A,  B  are  said  to  be  independent  if  knowing  that  one  will   occur  (or  has  occurred)  does  not  change  the  probability  that  the  other   occurs.  In  probability  notation  this  can  be  expressed  as  P(A|B)  =   P(A).           Sampling  with  and  without  Replacement   Definitions:  A  sample  is  drawn  with  replacement  if  individuals  are  returned   to  the  eligible  pool  for  each  selection.  A  sample  is  drawn  without   replacement  if  sampled  individuals  are  not  eligible  for  subsequent   selection.     -­‐   If  a  sample  is  drawn  from  a  very  large  population,  the  distinction   between  sampling  with  and  without  replacement  becomes   unimportant.     -­‐   Sometimes  students  confuse  the  mutually  exclusive  with   independence.  The  definition  for  two  events  to  be  disjoint  (mutually   exclusive)  was  based  on  a  SET  property.   ▯     -­‐   The  definition  for  two  events  to  be  independent  is  based  on  a   PROBABILITY  property.     -­‐   You  need  to  check  if  these  definitions  hold  when  asked  to  assess  if   two  events  are  disjoint,  or  if  two  events  are  independent.  


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