If two events are statistically independent of each other, the occurrence of one event has no influence on the occurrence of the other. Therefore, events A and B are independent if and only if P (A, B) = P (A) P (B). The probability of joint occurrence of K independent events can be generalized as
K
= P(A1) X P(A2) x-.-x P(Ak) = П P(Ak)
k=1
It should be noted that the mutual exclusiveness of two events does not, in general, imply independence, and vice versa, unless one of the events is an impossible event. If the two events A and B are independent, then A, A7, B, and B’ all are independent, but not necessarily mutually exclusive, events.
Example 2.2 Referring to Example 2.1, the probabilities that tributaries 1 and 2 overflow during a major storm event are 0.5 and 0.4, respectively. For simplicity, assume that the occurrences of overflowing in the two tributaries are independent of each other. Determine the probability of at least one tributary overflowing in a major storm event.
Solution Use the same definitions for events E1 and E2. The problem is to determine P(E1 U E2) by
P(E1 U E2) = P(E1) + P(E2) — P(E1, E2)
Note that in this example the probability of joint occurrences of both tributaries overflowing, that is, P(E1, E2), is not given directly by the problem statement, as in Example 2.1. However, it can be determined from knowing that the occurrences of
overflows in the tributaries are independent events, according to Eq. (2.5), as
P(Ex, E2) = P(E1)P(E2) = (0.5)(0.4) = 0.2
Then the probability that at least one tributary would overflow during a major storm event is
P(Ei U E2) = P(Ei) + P(E2) — P(Ex, E2) = 0.5 + 0.4 — 0.2 = 0.7
