Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

2.9 CONDITIONAL PROBABILITY

In many applications, information is often known about some events of an experiment that can influence the probability of other events of interest. It is possible to incorporate this prior information into the calculation of probabilities. Consider two events A and B in the probability space . If it is known that B has occurred, then the probability of A is usually different from that without this knowledge (though not necessarily). The notation P(A|B) is used to denote the probability of A given B (also stated as given B is true). The following is a simple example for which it easy to compute conditional probabilities.

Example 2.38. For the experiment of tossing two fair coins, let A be the event that both coins are heads, and B the event that at least one coin is heads. It is easy to interpret the conditioning event (i.e., knowing this additional information) as causing the sample space to be reduced from four elements to three elements. Event B causes the “new” sample space to be {HH, HT, TH} because TT is excluded, and thus we immediately have P(A|B) = 1/3. We can also compute P(B|A), though this probability turns out to be trivial. The reduced sample space is simply {HH}, from which it is clear that B (at least one H) is always true: P(B|A) = 1.

P(A|B) is a conditional probability. Although conditional probabilities were easy to calculate in Example ...