Conditional probability

   In modeling more complicated phenomena we may want to use different probabilities under different circumstances. For instance, in a modified blind search for the minimum of a non-negative function, the randomization strategy might be different when we already made some progress, and it might be different when we are ``stuck" in a non-optimal location. Thus we may want to consider probabilities of the same event A (say, hitting a maximum) under different conditions B.

To formalize this idea, suppose B is an event such that . The last condition merely says that B is an event that does have some chance of occurring. Conditional probability of event A given event B is denoted by . It is defined as

Conditional probability  satisfies the axioms of probability, and if A,B are disjoint. In particular, , .

The easiest way to find by simulations is to repeatedly simulate the complete experiment, discarding all the outcomes except the ones resulting in B.

 

You should notice that it takes forever to simulate events that happen rarely. Section indicates one possible way out of this difficulty.

• Properties of conditional probability
• Sequential experiments