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Probability
Questions ,[object Object],[object Object],[object Object]
“ frequentist” approach ,[object Object],[object Object],[object Object],[object Object],[object Object]
Bayesian approach ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
basic concepts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],basic concepts (cont.)
[object Object],[object Object],[object Object],[object Object],[object Object],basic concepts (cont.)
discrete probabilities 0 .25 .5 p HH TT HT
continuous probabilities total area under curve = 1 but the probability of any  single  value = 0   interested in the probability assoc. w/  intervals 0 .1 .2 p 0 .1 .2 p
independent events ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object]
conditional probability ,[object Object],[object Object]
e.g. ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],conditional probability (cont.)
Bayes Theorem ,[object Object]
application ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],25% jar 50% of bowls 80% of jars undec. 75% 50% of bowls 20% of jars ?? dec. bowl
Binomial theorem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
where n! = n*(n-1)*(n-2)…*1   (where n is an integer) 0!=1
misc. useful derivations from BT ,[object Object],[object Object],[object Object],[object Object],[object Object]
binomial distribution ,[object Object],[object Object],[object Object]
probability density function (PDF) ,[object Object]
ex: coin toss ,[object Object],[object Object],[object Object]
coin toss (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],“probability of  k  successes in  n  trials where the probability of success on any one trial is  p” HHH 3 H TT (THT,TTH) 1 HH T (HTH, THH) 2 TTT 0 k
 
practical applications ,[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],example
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
… 15 4 3 2 1 0 k 0.00 0.00 0.03 0.13 0.37 0.46 P(15,k,.05)
[object Object],[object Object],[object Object]
[object Object],[object Object]
What if wasters existed at a higher proportion than 5%??
so, how big should samples be? ,[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Pre-Dynastic cemeteries in Upper Egypt
[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object]
0.103 0.055 0.20 6 50 0.048 0.030 0.20 5 50 0.018 0.013 0.20 4 50 0.006 0.004 0.20 3 50 0.001 0.001 0.20 2 50 0.000 0.000 0.20 1 50 0.000 0.000 0.20 0 50 cumP P(n,k,p) p k n
 
[object Object],[object Object]

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4 probability

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  • 6.
  • 7.
  • 8. discrete probabilities 0 .25 .5 p HH TT HT
  • 9. continuous probabilities total area under curve = 1 but the probability of any single value = 0  interested in the probability assoc. w/ intervals 0 .1 .2 p 0 .1 .2 p
  • 10.
  • 11.
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  • 13.
  • 14.
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  • 20. where n! = n*(n-1)*(n-2)…*1 (where n is an integer) 0!=1
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  • 26.  
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  • 30.
  • 31. … 15 4 3 2 1 0 k 0.00 0.00 0.03 0.13 0.37 0.46 P(15,k,.05)
  • 32.
  • 33.
  • 34. What if wasters existed at a higher proportion than 5%??
  • 35.
  • 36.
  • 37.
  • 38.
  • 39.
  • 40.
  • 41.
  • 42.
  • 43.
  • 44.
  • 45. 0.103 0.055 0.20 6 50 0.048 0.030 0.20 5 50 0.018 0.013 0.20 4 50 0.006 0.004 0.20 3 50 0.001 0.001 0.20 2 50 0.000 0.000 0.20 1 50 0.000 0.000 0.20 0 50 cumP P(n,k,p) p k n
  • 46.  
  • 47.