conditioning and independence conditional probabilities and conditional distributions are used to model additional information on random experiment independence is used to model lack of such information conditional probability conditional probability suppose some event ohm occurs given this factevent will occur if and only if occursand the relative chance of occurring is therefore [ ]/ [ ]provided [ this leads to the definition of the conditional probability of given bp[ bp[ bp[bif [ ( the above definition breaks down if [ such conditional probabilities must be treated with more care [ three important consequences of the definition of conditional probability areproduct rule product rulefor any sequence of events an [ an [ [ [ [an an- ]( using the abbreviation ak : ak law of total probability law of total probabilityif {bi forms partition of ohm (that isbi and bi ohm)then for any event [ap[ bi [bi ( bayesrule bayesrulelet {bi form partition of ohm thenfor any event with [ [ [ [ ap [ bi [bi independent events ( independence two events and are said to be independent if the knowledge that has occurred does not change the probability that occurs that isab independent [ bp[asince [ bp[bp[ ]an alternative definition of independence is ab independent [ bp[ap[bthis definition covers the case where [ and can be extended to arbitrarily many eventsevents are said to be (mutuallyindependent if for any and any choice of distinct indices ik [ai ai aik [ai [ai [aik