L02.4 Conditional Probabilities Obey the Same Axioms
I now want to emphasize an important point. Conditional probabilities are just the same as ordinary probabilities applied to a different situation. They do not taste or smell or behave any differently than ordinary probabilities. What do I mean by that? I mean that they satisfy the usual probability axioms. For example, ordinary probabilities must also be non-negative. Is this true for conditional probabilities? Of course it is true, because conditional probabilities are defined as a ratio of two probabilities. Probabilities are non-negative. So the ratio will also be non-negative, of course as long as it is well-defined. And here we need to remember that we only talk about conditional probabilities when we condition on an event that itself has positive probability. How about another axiom? What is the probability of the entire sample space, given the event B? Let's check it out. By definition, the conditional probability is the probability of the intersection of the ...