12 4 8 12 4 8 12 Midnight Noon Midnight
FIGURE 1.1 A typical variation of sewage flow.
FIGURE 1.1 A typical variation of sewage flow.
FIGURE 1.3 Definition of event Z as "Going from A to B."
The path through E is an event or happening favorable to the occurrence of event Z. The other paths that a traveler could take to reach B are C, F, G, H, and I. Thus, the occurrence of any of these event paths will cause the occurrence of event Z; the occurrence of Z does not, however, mean that all of the event paths E, C, F, G, H, and I have occurred, but that at least one of them has occurred. These events are all said to be favorable to the occurrence of event Z.
The paths D, J, and K are events unrelated to Z; if the traveller chooses these paths she or he would never reach the destination point B. The events are not favorable to the occurrence of Z. All the events both favorable and not favorable to the occurrence of a given event, such as Z, constitute an event space of a particular domain. This particular domain space is called a probability space.
Addition rule of probability. Now, what is the probability that one event or the other will occur? The answer is best illustrated with the help of the Venn diagram, an example of which is shown in Figure 1.4, for the events A and B. There is D, which contains events from A and B; it is called the intersection of A and B, designated as A n B. This intersection means that D has events or results coming from both A and B. C has all its events coming from A, while E has all its events coming from B.
The sum of the events in A and B constitutes the union of A and B. This is written as A U B. From the figure,
where the subtraction comes from the fact that when A and B "unite," they each contribute to the events at the intersection part of the union (D). This part counted the intersection events twice; thus, the other "half" must be subtracted. The union of A and B is the occurrence of the event: event A or event B has occurred—not event A and event B have occurred. The event, event A and event B, have occurred is the intersection mentioned previously.
From Equation (1.1), the probability that one event or the other will occur can now be answered. Specifically, what is the probability that the one event A or the other event B will occur? Because the right side of the previous equation is equal to the left side, the probability of the right side must be equal to the probability of the left side. Or,
Prob (A U B) = Prob (A) + Prob (B) - Prob (A n B) (1.2)
where Prob stands for probability. Equation (1.2) is the probability that one event or the other will occur—the addition rule of probability.
Multiplication rule of probability. The intersection of A and B means that it contains events favorable to A as well as events favorable to B. Let the number of these events be designated as N(A n B). Also, let the number of events of B be designated as N(B). Then the expression
is the probability of the intersection with respect to the event B.
In the previous formula, event is synonymous with unit event. Unit events are also called outcomes. Probability values are referred to the total number of unit events or outcomes in the probability space, which would be the denominator of the above equation. As shown, however, the denominator of the above probability is referred to N(B). N(B) is smaller than the total number of unit events in the domain space; thus, it is called a reduced probability space. Because the reference probability space is that of B and because N(A n B) is equal to the number of unit events of A in the intersection, the previous equation is called the conditional probability of A with respect to B designated as Prob(A|B), or
Let Z designate the total number of unit events in the domain probability space in which event A is a part as well as event B is a part. Divide the numerator and the denominator of the above equation by Z Thus,
The numerator of the previous equation is the intersection probability Prob(A n B) and the denominator is Prob(B). Substituting and performing the algebra, the following equation is produced:
Equation (1.6) is the multiplication rule of probability. If the reduced space is referred to A, then the intersection probability would be
In the multiplication rule, if one event precludes the occurrence of the other, the intersection does not exist and the probability is zero. These events are mutually exclusive.
If the occurrence of one event does not affect the occurrence of the other, the events are independent of each other. They are independent, so their probabilities are also independent of each other and Prob(A |B) becomes Prob(A) and Prob(A |B) becomes Prob(B).
Using the intersection probabilities, the addition rule of probability becomes Prob(A U B) = Prob(A) + Prob(B) - Prob(A|B)Prob(B) (1.8)
One of the values that is often determined is the value equaled or exceeded. The probability of a value equaled or exceeded may be calculated by the application of the addition rule of probability. The phrase "equaled or exceeded" denotes an element equaling a value and elements exceeding the value. Therefore, the probability that a value is equaled or exceeded is by the addition rule,
But there are 1, 2, 3,... y of the elements exceeding the value. Also, for mutually exclusive events, the intersection probability is equal to zero. Thus,
Substituting Equation (1.10) into Equation (1.9), assuming mutually exclusive events,
Prob (value equaled or exceeded)
= Prob(value equaled) + Prob(value1 exceeding)
+ Prob(value2 exceeding) + — + Prob(valuey exceeding) (1.11)
1.1.3 Derivation of Probability from Recorded Observation
In principle, to determine the probability of occurrence of a certain event, the experiment to determine the total number of unit events or outcomes for the probability space should be performed. Then the probability of occurrence of the event is equal to the number of unit events favorable to the event divided by the total possible number of unit events. If the number of unit events favorable to the given event is n and the total possible number of unit events in the probability space is Z, the probability of the event, Prob(E), is
Prob(value equaled or exceeded)
= Prob(value equaled) + Prob(value exceeded) - Prob ( value equaled n value exceeded)
Prob (value exceeded)
= Prob(valuel exceeding) + Prob(value2 exceeding) + — + Prob(value y exceeding)
In practical situations, either the determination of the total number of Js is very costly or the total number is just not available. Assume that the available Z is Zavan, then the approximate probability, Prob(E)approx, is
The use of the previous equation, however, may result in fallacy, especially if Zavau is very small. n can become equal to Zotai making the probability equal to 1 and claiming that the event is certain to occur. Of course, the event is not certain to occur, that is the reason why we are using probability. What can be claimed with correctness is that there is a high degree of probability that the event will occur (or a high degree of probability that the event will not occur). Because there is no absolute certainty, in practice, a correction of 1 is applied to the denominator of Equation (1.13) resulting in
Take note that for large values of ZavaU the correction 1 in the denominator becomes negligible.
To apply Equation (1.14), recorded data are arranged into arrays either from the highest to the lowest or from the lowest to the highest. The number of values above a given element and including the element is counted and the probability equation applied to each individual element of the array. Because the number of values above and at a particular element is a sum, this application of the equation is, in effect, an application of the probability of the union of events. The probability is called cumulative, or union probability. After all union probabilities are calculated, an array of probability distribution results. This method is therefore called probability distribution analysis. This method will be illustrated in the next example.
Example 1.1 In a facility plan survey, data for Sewer A were obtained as follows:
1 2900 8 4020
2 3028 9 3675
3 3540 10 3785
4 3300 11 3459
5 3700 12 3200
6 4000 13 3180
7 3135 14 3644
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