1 Probability of event A conditional on event B occurring E R Morey ConditionalProbability tex September 8 2010 Let Pr A jB Probability of event A conditional on event B occurring assuming Pr B 0 if Pr B 0 Pr A jB is not de ned Consider the following f x y 0 x 100 0 y 100g Now consider two sets A f x y 20 x 50 40 y 60g and B f x y 0 x 40 50 y 100g Note that A and B are each events Note that in this example these two event sets partially intersect We could ask the probability of a draw from A given that we are drawing from set B this is a conditional probability Consider other examples where A B or A B Draw some Venn diagrams 1 De nition 1 of conditional probability 1 Pr A jB if Pr B 0 Read Pr AB Pr A B Pr AB Pr B Pr A and B Why can t the conditional probability be de ned as Pr A jB Pr AB Since there is nothing special about the names A and B 2 Pr B jA Pr AB Pr A if Pr A 0 One can rearrange 1 and 2 to obtain 1a and 2a Pr AB Pr A jB Pr B and Pr BA Pr B jA Pr A Note that Pr AB Pr BA Combining 1a and 2a one obtains 3 Pr AB Pr A jB Pr B Pr B jA Pr A Pr BA What is the intuition behind de ning conditional probability as we did above In conditional probability the sample space is e ectively reduced to B that is the probability that A will happen given that ones lives in a world where B prevails So Pr A jB is the probability that both A and B occur Pr AB as a proportion of the probability of B Pr B 2 Note the following which following from 2 and 3 divide 3 by Pr A Pr B jA Pr BA Pr AB Pr A Pr A Pr A jB Pr B Pr A This says that if one knows Pr A jB Pr B and Pr A one can determine Pr B jA What is this result called Bayes theorem Bayes theorem is quite useful I have used it in my research on numerous occasions For example assume that one has determined that there are 4 classes types of individuals and that one has determined the probability that an individual will choose alternative 1 conditional on being a member of class III Pr 1 jIII determined the probability of answering 1 Pr 1 and determined the probability of being in class III Pr III One can use Bayes theorem to determine the probability that an individual is in class III given that they chose alternative 1 Pr III j1 Work it out Applying Bayes theorem Pr III j1 Pr 1 jIII Pr III Pr 1 3 There are many problems in the rst review set about Bayes theorem Bayes theorem is amazing Look at the vegetarian dog problem and the gubergomer problem 4 1 1 When are events A and B independent Not independent Note that at this point we are talking about events not random variables Start by considering a case where Pr A 0 and Pr B 0 but A B so Pr AB 0 If these assumptions hold Pr A jB Pr B jA 0 that is the two events are mutually exclusive Draw a Venn diagram In this case are A and B independent NO One event happening precludes the other event from happening they are dependent When do we say A and B are independent De nition 2 A and B are independent i the following are true each is a di erent way of saying the same thing Pr AB Pr A B Pr A Pr B Pr A jB Pr A if Pr B 0 Pr B jA Pr B if Pr A 0 5 Note that these are three equivalent statements each implies the other two Hopefully you can show this There is only one piece of information in the three equations not three The following can be deduced from the de nition of independence 1 Pr A 0 Pr B 0 and A B A and B not independent this was my example above they are dependent Draw a Venn diagram to convince yourself this is true I typically get confused on this issue if two sets have no intersection it is easy to wrongly conclude that they are independent they don t tough each other In the terminology of necessary and su cient 1 says Pr A 0 Pr B 0 and A B is su cient for A and B not independent In terms of the arrow 1 Pr A 0 Pr B 0 and A B A and B not independent Another way of expressing this is not A and B not independent not Pr A 0 Pr B 0 and A B Which is equivalent to A and B are independent not Pr A 0 Pr B 0 and A B 1 To say that x is su cient for y is equivalent to saying x implies y the existence of x guarentees y It follows by the rules of logit that if x implies y then noty implies notx 6 The following can also be deduced 2 Pr A 0 Pr B 0 and A and B are independent A B 6 In the terminology of necessary and su cient 2 says Pr A 0 Pr B 0 and A and B are independent is su cient for A B 6 Another way of expressing this is not A B 6 not Pr A 0 Pr B 0 and A and B are independent Which is equivalent to A B not Pr A 0 Pr B 0 and A and B are independent Think about how 1 and 2 are di erent They are not equivalent statements 7 Then think about the distinction between the following two statements Pr A 0 Pr B 0 and A B dent this is 1 above A and B are dependent not indepen and A B is consistent with A and B being independent Both of these statements are correct statements But if A B A and B are independent only if one of the sets is empty P A P B 0 Notice the rst statement says that when A B and some other conditions hold A and B are dependent while the second statement notes that A B and A and B can be independent If two non empty sets are independent they must have common elements a non empty intersection Does it go the other way Does A B 6 imply that A and B are independent No 8 Independence of n events is more complicated than independence of 2 events De nition 3 A1 A2 A3 and A4 are independent i all of the following are true a Pr Ai Aj Pr Ai Pr Aj 8i and j i 6 …