Probability
an event in a single trial. It also describes the long-run relative frequency of occurrence of
getting heads in many trials (out of 50 coin flips, we expect to average 25 heads).
Probabilities can also be based on data, expert judgment, or a combination of both.
Past data on weather and climate, on project completion times and costs, and on highway
traffic is combined with expertjudgment to forecast future events. These examples can be
important in engineering economy.
Another example based on long-run relative frequency is the PW of a flood-protection
dam that depends on the probabilities of different-sizedfloods over many years. This would
be based on data about past floods and would include many years of potential flooding.
An example of a single event that may be estimated by expert judgment is the probability
of a successful outcome for a research and development project, which will determine
its PW.
All the data in an engineering economy problem may have some level of uncertainty.
However,smalluncertaintiesmay be ignored,so that moreanalysiscan be done withthe.
large uncertainties. For example, the price of an off-the-shelf piece of equipment may vary
by only ::1::5%.The price could be treated as a known or deterministic value. On the other
hand, demandover the next 20 years will havemore uncertainty.Demandshould be 3l\alyzed
as a random or stochastic variable. We should establish probabilities for different values of
demand.
There are also logical or mathematical rules for probabilities. If an outcome can never
happen, then the probability is O.If an outcome will certainly happen, then the probability
is 1, or 100%. This means that probabilities cannot be negative or greater than 1; in other
words, they must be within the interval [0, 1], as indicated shortly in Equation 10-2.
Probabilities are defined so that the sum of probabilities for all possible outcomes is
1 or 100% (Equation 10-3). Summing the probability of 0.5 for a head and 0.5 for a tail
leads to a total of 1 for the possible outcomes from the coin flip. An explorationwell drilled
in a potential oil field will have three outcomes (dry hole, noncommercial quantities, or
commercial quantities) whose probabilities will sum to one.
Equations 10-2 and 10-3 can be used to check that probabilities are valid. If the prob-
abilities for all but one outcome are known, the equations can be used to find the unknown
probability for that outcome (see Example 10-5).o ~ Probability ~ 1 (10-2)
L P(outcomej) =1, where there areKoutcomes (10-3)
j=l toKL
rl
L
In a probability course many probability distributions, such as the normal, uniform,
and beta are presented. These continuous distributions describe a large population of data.
However,for engineering economy it is more common to use 2 to 5 outcomes with discrete
probabilities--even though the 2 to 5 outcomes only represent or approximate the range of
possibilities.
This is done for two reasons. First, the data often are estimated by expert judgment, so
that using 7 to 10 outcomes would be false accuracy. Second, each outcome requires more
analysis. In most cases the 2 to 5 outcomesrepresents the best trade-offbetweenrepresenting
the range of possibilities and the amount of calculation required. Example 10-5 illustrates
these calculations.-- --- --