2.1.1 Basic axioms of probability
The three basic axioms of probability computation are (1) nonnegativity: P (A) > 0, (2) totality: P (S) = 1, with S being the sample space, and (3) additivity: For two mutually exclusive events A and B, P(A U B) = P(A) + P(B).
As indicated from axioms (1) and (2), the value of probability of an event occurring must lie between 0 and 1. Axiom (3) can be generalized to consider K mutually exclusive events as
P (A1 U A2 U-.-U Ak ) = p( U aA =£) P (Ak) (2.2)
‘vk=1 ‘ k=1
An impossible event is an empty set, and the corresponding probability is zero, that is, P(0) = 0. Therefore, two mutually exclusive events A and B have zero probability of joint occurrence, that is, P (A, B) = P (0) = 0. Although the probability of an impossible event is zero, the reverse may not necessarily be true. For example, the probability of observing a flow rate of exactly 2000 m3/s is zero, yet having a discharge of 2000 m3/s is not an impossible event.
Relaxing the requirement of mutual exclusiveness in axiom (3), the probability of the union of two events can be evaluated as
P(A U B) = P(A) + P(B) — P(A, B) (2.3)
which can be further generalized as
/ K N K
W ^AkJ = £ P (Ak) — ££P ( a Aj •
^ ‘ k = 1 i < j
+ EEE P(At, A, Ak) -■ ■ ■ + (-1)KP(A1, A2,…, Ak)
i < j < k
(2.4)
If all are mutually exclusive, all but the first summation term on the right-hand side of Eq. (2.3) vanish, and it reduces to Eq. (2.2).
Example 2.1 There are two tributaries in a watershed. From past experience, the probability that water in tributary 1 will overflow during a major storm event is 0.5, whereas the probability that tributary 2 will overflow is 0.4. Furthermore, the probability that both tributaries will overflow is 0.3. What is the probability that at least one tributary will overflow during a major storm event?
Solution Define Ei = event that tributary i overflows for i = 1, 2. From the problem statements, the following probabilities are known: P(E]_) = 0.5, P(E2) = 0.4, and P(E1, E2) = 0.3.
The probability having at least one tributary overflowing is the probability of event E1 or E2 occurring, that is, P(E1 U E2). Since the overflow of one tributary does not preclude the overflow of the other tributary, E1 and E2 are not mutually exclusive. Therefore, the probability that at least one tributary will overflow during a major storm event can be computed, according to Eq. (2.3), as
P(E1 U E2) = P(E1) + P(E2) — P(E1, E2) = 0.5 + 0.4 — 0.3 = 0.6
