The conditional probability is the probability that a conditional event would occur. The conditional probability P (A | B) can be computed as
P(AI B) = PAA <2.6)
in which P (A | B) is the occurrence probability of event A given that event B has occurred. It represents a reevaluation of the occurrence probability of event A in the light of the information that event B has occurred. Intuitively, A and B are two independent events if and only if P (A | B) = P (A). In many cases it is convenient to compute the joint probability P (A, B) by
P (A, B) = P (B) P (A | B) or P (A, B) = P (A) P (B | A)
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The probability of the joint occurrence of K dependent events can be generalized as
Example 2.3 Referring to Example 2.2, the probabilities that tributaries i and 2 would overflow during a major storm event are 0.5 and 0.4, respectively. After examining closely the assumption about the independence of overflow events in the two tributaries, its validity is questionable. Through an analysis of historical overflow events, it is found that the probability of tributary 2 overflowing is 0.6 if tributary i overflows. Determine the probability that at least one tributary would overflow in a major storm event.
Solution Let Ei and E2 be the events that tributary i and 2 overflow, respectively. From the problem statement, the following probabilities can be identified:
P(Ei) = 0.5 P(E2) = 0.4 P(E2 | Ei) = 0.6
in which P(E2 | Ei) is the conditional probability representing the likelihood that tributary 2 would overflow given that tributary i has overflowed. The probability of at least one tributary overflowing during a major storm event can be computed by
P(Ei U E2) = P(Ei) + P(E2) — P(Ei, E2)
in which the probability of joint occurrence of both tributaries overflowing, that is, P(E1, E2), can be obtained from the given conditional probability, according to Eq. (2.7), as
P(E1, E2) — P(E2 | E1)P(E1) — (0.6)(0.5) — 0.3
The probability that at least one tributary would overflow during a major storm event can be obtained as
P(E1 U E2) — P(E1) + P(E2) — P(E1, E2) — 0.5 + 0.4 — 0.3 — 0.6
