1  

Davis  -­‐  Chp.  2  –  Elementary  Statistics  

Probability  (pp.  11-­‐24)  

Binomial  Combinations  -­‐  Number  of  combinations  of  n  items  taken  r  at  a  time:  

!! =

!!! − ! ! !!  

Binomial  Distribution  –  The  classic  coin  toss  example:  r  successes  will  occur  in  n  trials  

! =!! (1− !)!!!!! =

!!! − ! ! !! (1− !)

!!!!!  

Negative  Binomial  –  “How  many  wells  do  we  have  to  drill  before  a  r  discoveries  are  made?”  

! =(! + ! + 1)!! − 1 ! !! (1− !)

!!!  

Sampling  with/without  Replacement  

Hypergeometric  Probability  Distribution  –  “What  are  the  chances  of  x  discoveries  out  of  n  drill  holes  if  N  prospects  contain  S  reservoirs?”  

! =!!

!!!!!!!!

=

!!! − ! ! !!

(! − !)!(! − !)− (! − !) ! (! − !)!

!!! − ! !!!

 

Mutually  Exclusive  Events  and  Additive  Rule  of  Probability  

When  only  a  discrete  number  of  outcomes  are  possible  and  they  are  all  mutually  exclusive,  then  

! !  !"  !  !"  ! = ! ! + ! ! + !(!)  

Independent  Events  and  the  Multiplicative  Rule  of  Probability  

Conditional  Probabilities    -­‐  If  harmonic  tremors  occur  from  magma  movement  in  a  volcano  AND  eruptions  follow  magma  movement  in  a  volcano,  then  there’s  a  relationship  between  harmonic  tremors  and  eruptions,  so  the  probability  of  a  tremor  AND  an  eruption  occurring  is  not  equal  to  the  probability  of  a  tremor  X  the  probability  of  an  eruption.  

Bayes’  Theorem  –  The  joint  probability  that  both  events  A  and  B  occur  is  equal  to  the  probability  that  B  will  occur  given  that  A  has  already  occurred  times  the  probability  that  A  will  occur.  

Bayes’  Basic  Equation:  

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!(!,!) = ! !|! !(!)  

Converse  of  Bayes’  Basic  Equation:  

!(!,!) = ! !|! !(!)  

Equating  Bayes’  Basic  Equation  and  its  converse:  

! !|! !(!) = ! !|! !(!)  

Solving  for  the  probability  of  B’s  occurrence,  given  A’s  prior  occurrence  gives  Bayes’  Theorem  in  its  common  form:  

! !|! =! !|! !(!)

!(!)  

!(!) = ! !|!! !(!!)!

!!!  

Bayes’  Theorem  for  individual  events  that  are  conditionally  related  to  A:  

! !!|! =! !|!! !(!!)! !|!! !(!!)!

!!!  

Nice  Example  illustrating  the  use  of  Bayes’  Theorem  provided  –  determination  of  likely  source  area  of  a  fossil  found  downstream  of  a  tributary.  

 

Continuous  Random  Variables  (pp.  25-­‐29)  

Increasing  number  of  coin  flips  narrows  the  bin  widths  on  the  histogram,  decreasing  the  probability  that  any  one  result  (e.g.  21  out  of  50  flips  will  be  heads)  will  occur.  

Experimental  Error  and  Confounded  Sources  of  Variation  

Normal  Distribution  

Population  and  Sample  

 

Statistics  (pp.  29-­‐33)  

Parameters  describe  characteristics  of  population  distributions,  whereas  Statistics  describe  characteristics  of  sample  distributions.  

Frequency  Histogram  vs.  Relative  Frequency  Histogram  

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Cumulative  Plot  

Quantiles,  Percentiles,  Deciles,  Quartiles  

Box-­‐and-­‐Whisker  Plots  –  show  population  distributions  in  a  cartoon-­‐like  fashion  

 

Summary  Statistics  (pp.  33-­‐39)