4/24/2012 2:00:00 AM Lecture 9 Modeling Variation With Probability Part 3 Review Rules 5a 5b and 5c 5a) P(A|B)=P(A and B) /P(B) 5b) rearranging fives 2 forms P(A and B)=P(B) P(A|B) P(A and B) =P(A) P(B|A) 5c) If A and B are independent events then 5b reduces to P(A and B) =P(A) P(B)  we use 5c when events are independent of there is no association Determining Independence or Dependence  if a person is selected at random is the characteristic that a person is a democrat independent of the characteristic that he/she is liberal? Dem Rep Other Total Liberal 306 20 198 530 Moderate 279 134 3322 735 Conservative 104 309 180 593 Total 689 469 700 1858 Is P(A|B)=P(A)? P(Democrat) =689 / 1858 -.3708 P(Democrat| Liberal) =306 / (306 + 26 + 198) = .577  DEPENDENTAnother Way (an alternative way) P(A and B) = P(A) P(B) P(Democrat)=689/1858 =.3708 P(Liberal)=530/1858 - .2853 P(Democrat and Liberal)= 306/1858 =.1646 P(Democrat) * P(Liberal) =.1058 DEPENDENT Sequences Again  if you cant verify independence than you must use rule 5b P(A and B) = P(A) P(B|A)  the key is to have Information of P(B|A) Empirical Probabilities  useful when we are not sure what assumption we need to make to find theoretical probabilities  useful when theoretical probability Is to complex Summary of a Situation Step 1: identify the actions that make up a single trial Step2: decide how to represent these outcomes Step 3: describe the outcomes you will record Step 4: describe outcomes of interest Step 5: preform the simulation with many repetitions Step 6: calculate the proportion of trials in which the outcome of interest occurred The result is the Empirical probabilityThe Law of Large Numbers  if an experiment with a random outcome is repeated a large number of times the empirical provability of an event is likely to be clost to the true theoretical probability  the large the number of repetitions get the closer these probabilities will be to real world numbers LLN  if you surveyed 5000 Americans ages 18-60 to find out the proportions of who is currently unemployed the result will most likely be representative or close to the proportion of ALL Americans that are currently