Conditional probability
A conditional probability refers to the probability of observing an event A given that you have observed a separate event B. The mathematical shorthand for expressing this idea is:
P(A  B)
Imagine that A refers to "customer buys product A" and B refers to "customer buys product B". P(A  B) would then read as the "probability that a customer will buy product A given that they have bought product B." If A tends to occur when B occurs, then knowing that B has occurred allows you to assign a higher probability to A's occurrence than in a situation in which you did not know that B occurred.
More generally, if A and B systematically covary in some way, then P(A  B) will not be equal to P(A). Conversely, if A and B are independent events, then P(A  B) would be expected to equal P(A).
The need to compute a conditional probability thus arises any time you think the occurence of some event has a bearing on the probability of another event's occurring.
The most basic and intuitive method for computing P(A  B) is the set enumeration method. Using this method, P(A  B) can be computed by counting the number of times A and B occur together {A & B} and dividing by the number of times B occurs {B}:
P(A  B) = {A & B} / {B}
If you observe that 12 customers to date bought product B and of those 12, 10 also bought product A, then P(A  B) would be estimated at 10/12 or 0.833. In other words, the probability of a customer buying product A given that they have purchased product B can be estimated at 83 percent by using a method that involves enumerating relative frequencies of A and B events from the data gathered to date.
You can compute a conditional probability using the set enumeration method with the following PHP code:
Listing 1. Computing conditional probability using set enumeration

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