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B–3 Probability and Relative Frequency 683
From the definition of probability, as given by Eq. (B–1), it is seen that all probability
functions have the property
0 … P(A) … 1
(B–2)
where P(A) = 0 if the event A is a null event (never occurs) and P(A) = 1 if the event A is a
sure event (always occurs).
Joint Probability
DEFINITION.
The probability of a joint event, AB, is
P(AB) = lim
n: q an AB
n b
(B–3)
where n AB is the number of times that the event AB occurs in n trials.
In addition, two events, A and B, are said to be mutually exclusive if AB is a null event, which
implies that P(AB) K 0.
Example B–1 EVALUATION OF PROBABILITIES
Let event A denote an auto accident blocking a certain street intersection during a 1-min interval.
Let event B denote that it is raining at the street intersection during a 1-min period. Then, event
E = AB would be a blocked intersection while it is raining, as evaluated in 1-min increments.
Suppose that experimental measurements are tabulated continuously for 1 week, and it is
found that n A = 25, n B = 300, n AB = 20, and there are n = 10,080 1-min intervals in the week of
measurements. (n A = 25 does not mean that there were 25 accidents in a 1-week period, but that
the intersection was blocked for 25 1-min periods because of car accidents; this is similarly
true for n B and n AB .) These results indicate that the probability of having a blocked intersection is
P(A) = 0.0025, the probability of rain is P(B) = 0.03, and the probability of having a blocked
intersection and it is raining is P(AB) = 0.002.
The probability of the union of two events may be evaluated by measuring the compound
event directly, or it may be evaluated from the probabilities of simple events as given
by the following theorem:
THEOREM.
Let E = A + B; then,
P(E) = P(A + B) = P(A) + P(B) - P(AB)
(B–4)
Proof. Let the event A only occur n 1 times out of n trials, the event B only occurs n 2
times out of n trials, and the event AB occur n AB times. Thus,
P(A + B) = lim an 1 + n 2 + n AB
b
n: q n
= lim an 1 + n AB
b + lim
n: q n
an 2 + n AB
b - lim
n: q n
n b
n: q an AB